# Full text of "Projective Geometry Volume I"

## See other formats

THE TEXT IS LIGHT IN THE BOOK CARNEGIE INSTITUTE OF TECHNOLOGY LIBRARY PRESENTED BY Dr, Lloyd L* Dines PEOJECTIVE GEOMETRY BY OSWALD VEBLEN PROFESSOR OP MATHEMATICS, PRINCETON UNIVERSITY AND JOHN WESLEY YOUNG PROFESSOR OF MATHEMATICS, UNIVERSITY OF KANSAS VOLUME I GINN AND COMPANY BOSTON * NEW YORK CHICAGO LONDON ENTERED AT STATIONERS' HALL COPYRIGHT, 1910, BY OSWALD VEBLEN AND JOHN WESLEY YOUNG ALL RIGHTS RESERVED 810.9 fltfrenteum GINN AND COMPANY PRO- PRIETORS BOSTON U.S.A. PREFACE Geometry, which had been for centuries the most perfect example of a deductive science, during the creative period of the nineteenth century outgrew its old logical forms. The most recent period has however brought a clearer understanding of the logical foundations of mathematics and thus has made it possible for the exposition of geometry to resume the purely deductive form. But the treatment in the books which have hitherto appeared makes the work of lay- ing the foundations seem so formidable as either to require for itself a separate treatise, or to be passed over without attention to more than the outlines. This is partly due to the fact that in giving the complete foundation for ordinary real or complex geometry, it is necessary to make a study of linear order and continuity, a study which is not only extremely delicate, but whose methods are those of the theory of functions of a real variable rather than of elemen- tary geometry. The present work, which is to consist of two volumes and is in- tended to be available as a text in courses offered in American uni- versities to upper-class and graduate students, seeks to avoid this difficulty by deferring the study of order and continuity to the sec- ond volume. The more elementary part of the subject rests on a very simple set of assumptions which characterize what may be called "general protective geometry." It will be found that the theorems selected on this basis of logical simplicity are also elemen- tary in the sense of being easily comprehended and often used. Even the limited space devoted in this volume to the foundations may seem a drawback from the pedagogical point of view of some mathematicians. To this we can only reply that, in our opinion, an adequate knowledge of geometry cannot be obtained without attention to the foundations. "We believe, moreover, that the abstract treatment is peculiarly desirable in protective geometry, because it is through the latter that the other geometric disciplines are most readily coordinated. Since it is more natural to derive iii iv PllEFAOE the geometrical disciplines associated with the names of Euclid, Descartes, Lobatchewsky, etc., from protective geometry than it is to derive projective geometry from one of them, it is natural to take the foundations of projective geometry as the foundations of all geometry. The deferring of linear order and continuity to the second vol- ume has necessitated the deferring of the discussion of the metric geometries characterized by certain subgroups of the general pro- jective group. Such elementary applications as the metric proper- ties of conies will therefore be found in the second volume. This will be a disadvantage if the present volume is to be used for a short course in which it is desired to include metric applications. But the arrangement of the material will make it possible, when the second volume is ready, to pass directly from Chapter VIII of the first volume to the study of order relations (which may them- selves be passed over without detailed discussion, if this is thought desirable), and thence to the development of Euclidean metric geometry. We think that much is to be gained pedagogioally as well as scientifically by maintaining the sharp distinction between the projective and the metric. 1 The introduction of analytic methods on a purely synthetic basis in Chapter VI brings clearly to light the generality of the set of assumptions used in this volume. What we call " general projective geometry " is, analytically, the geometry associated with a general number field. All the theorems of this volume are valid, not alone in the ordinary real and the ordinary complex projective spaces, but also in the ordinary rational space and in the finite spaces. The bearing of this general theory once fully comprehended by the student, it is hoped that he will gain a vivid conception of the organic unity of mathematics, which recent developments of postu- lational methods have so greatly emphasized. The form of exposition throughout the book has been condi- tioned by the purpose of keeping to the fore such general ideas as group, configuration, linear dependence, the correspondence be- tween and the logical interchangeability of analytic and synthetic methods, etc. Between two methods of treatment we have chosen the more conventional in all cases where a new method did nol to have unquestionable advantages. We have tried also to Jb'JbC.U.FAU.U V avoid in general the introduction of new terminology. The use of the word on in connection with duality was suggested by Pro- fessor Frank Morley. We have included among the exercises many theorems which in a larger treatise would naturally have formed part of the text. The more important and difficult of these have been accompanied by references to other textbooks and to journals, which it is hoped will introduce the student to the literature in a natural way. There has been no systematic effort, however, to trace theorems to their original sources, so that the book may be justly criticized for not always giving due credit to geometers whose results have been used. Our cordial thanks are due to several of our colleagues and stu- dents who have given us help and suggestions. Dr. H. H. Mitchell has made all the drawings. The proof sheets have been read in whole or in part by Professors Birkhoff, Eisenhart, and Wedderburn, of Princeton University, and by Dr. R. L. Borger of the University of Illinois. Finally, we desire to express to Ginn and Company our sincere appreciation of the courtesies extended to us. It is expected that the second volume will appear during the coming year. O. VEBLEN J. W. YOUNG- August, 1910 CONTENTS INTRODUCTION SECTION PAGE 1. Undefined elements and unproved propositions 1 2. Consistency, categoricalness, independence. Example of a mathematical science 2 3. Ideal elements in geometry 7 4. Consistency of the notion of points, lines, and plane at infinity 9 5. Protective and metric geometry 12 CHAPTER I THEOREMS OF ALIGNMENT AND THE PRINCIPLE OF DUALITY 6. The assumptions of alignment 15 7. The plane 17 8. The first assumption of extension 18 9. The three-space 20 10. The remaining assumptions of extension for a space of three dimensions . 24 11. The principle of duality 26 12. The theorems of alignment for a space of n dimensions 29 CHAPTER II PROJECTION, SECTION, PERSPECTIVITY. ELEMENTARY CONFIGURATIONS 13. Projection, section, perspectivity 34" 14. The complete n-point, etc 36 15. Configurations 38 16. The Desargues configuration 39 17. Perspective tetrahedra 43 18. The quadrangle-quadrilateral configuration 44 19. The fundamental theorem on quadrangular sets 47 20. Additional remarks concerning the Desargues configuration 51 CHAPTER III PROJEOTIVITIES OF THE PRIMITIVE GEOMETRIC FORMS OF ONE, TWO, AND THREE DIMENSIONS 21. The nine primitive geometric forms 55 22. Perspectivity and projectivity 56 23. The projectivity of one-dimensional primitive forms 59 vii viii CONTENTS SECTION PAGE 24. General theory of correspondence. Symbolic treatment ....... 64 25. The notion of a group ..................... 06 26. Groups of correspondences. Invariant elements and figures ..... 07 27. Group properties of pro jectivi ties ................ 08 28. Protective transformations of two-dimensional forms ........ 71 29. Protective collineations of three-dimensional forms ......... 75 CHAPTER IV HARMONIC CONSTRUCTIONS AND THE FUNDAMENTAL THEOREM OF PROJECTIVE GEOMETRY 30. The projectivity of quadrangular sets .............. 70 31. Harmonic sets ........................ 80 32. Nets of rationality on a line .................. 84 33. Nets of rationality in the plane ................. 86 34. Nets of rationality in space ................... 80 35. The fundamental theorem of projectivity ............. 03 36. The configuration of Pappus. Mutually inscribed and circumscribed tri- angles .......................... 08 37. Construction of projectivities on one-dimensional forms ....... 100 38. Involutions ......................... 102 39. Axis and center of homology .................. 103 40. Types of collineations in the plane ............... 106 CHAPTER V CONIC SECTIONS 41. Definitions. Pascal's and Brianchon's theorems .......... 100 42. Tangents. Points of contact .................. 112 43. The tangents to a point conic form a line conic ........... 116 44. The polar system of a conic ................. . 120 45. Degenerate conies ...................... 120 46. Desargues's theorem on conies ................. 127 47. Pencils and ranges of conies. Order of contact .......... 128 CHAPTER VI ALGEBRA OF POINTS AND ONJG-DIMENSIONAL COORDINATE SYSTEMS 48. Addition of points ...................... 141 49. Multiplication of points .................... 144 50. The commutative law for multiplication ............. 148 51. The inverse operations .................... 148 52. The abstract concept of a number system. Isomorphism ....... 149 53. Nonhomogeneous coordinates ................. 150 54. The analytic expression for a projectivity in a one-dimensional primitive form ........................... 152 55. Von Staudt's algebra of throws ................. 157 CONTENTS ix SECTION PAGE 56. The cross ratio 159 57. Coordinates in a net of rationality on a line , 1C2 58. Homogeneous coordinates on a line 163 59. Protective correspondence "between the points of two different lines . . 166 CHAPTEE VII COORDINATE SYSTEMS IN TWO- AND THEEE- DIMENSIONAL FORMS 60. Nonhomogeneous coordinates in a plane 169 61. Simultaneous point and line coordinates 171 62. Condition that a point be on a line 172 63. Homogeneous coordinates in the plane 174 64. The line on two points. The point on two lines 180 65. Pencils of points and lines. Projectivity 181 66. The equation of a conic 185 67. Linear transformations in a plane . * 187 68. Collineations between two different planes 190 69. Nonhomogeneous coordinates in space 190 70. Homogeneous coordinates in space 194 71. Linear transformations in space 199 72. Finite spaces 201 CHAPTER VIII PROJECTIVITIES IN ONJE- DIMENSIONAL FORMS 73. Characteristic throw and cross ratio 205 74. Projective projectivities 208 75. Groups of projectivities on a line 209 76. Projective transformations between conies 212 77. Projectivities on a conic 217 78. Involutions 221 79. Involutions associated with a given projectivity 225 80. Harmonic transformations 230 81. Scale on a conic 231 82. Parametric representation of a conic 234 CHAPTER IX GEOMETRIC CONSTRUCTIONS. INVARIANTS 83. The degree of a geometric problem 236 84. The intersection of a given line with a given conic 240 85. Improper elements. Proposition K 2 241 86. Problems of the second degree 245 87. Invariants of linear and quadratic binary forms 251 88. Proposition K n 254 89. Taylor's theorem. Polar forms 255 X. CONTENTS 90. Invariants and covariants of binary forms 257 91. Ternary and quaternary forms and their invariants 258 92. Proof of Proposition K n 200 CHAPTER X PROJECTIVE TRANSFORMATIONS OF TWO-DIMENSIONAL FORMS 93. Correlations between two-dimensional forms 202 94. Analytic representation of a correlation between two planes 200 95. General projective group. Representation by matrices 208 96. Double points and double lines of a collineation in a piano 271 97. Double pairs of a correlation 278 98. Fundamental conic of a polarity in a plane 282 99. Poles and polars with respect to a conic. Tangents 284 100. Various definitions of conies 285 101. Pairs of conies 287 102. Problems of the third and fourth degrees 294 CHAPTER XI FAMILIES OF LINES 103. The regains 21)8 104. The polar system of a regulus 300 105. Projective conies 304 106. Linear dependence of lines 1)11 107. The linear congruence 312 108. The linear complex 319 109. The Pliicker line coordinates 327 110. Linear families of lines 329 111. Interpretation of line coordinates as point coordinates in S 6 331 INDEX 335 PROJECTIVE GEOMETRY INTRODUCTION 1. Undefined elements and unproved propositions. Geometry deals with the properties of figures in space. Every such figure is made up of various elements (points, lines, curves, planes, surfaces, etc.), and these elements bear certain relations to each other (a point lies on a line, a line passes through a point, two planes intersect, etc.). The propositions stating these properties are logically interdependent, and it is the object of geometry to discover such propositions and to exhibit their logical interdependence. Some of the elements and relations, by virtue of their greater simplicity, are chosen as fundamental, and all other elements and relations are defined.in terms of them. Since any defined element or relation must be defined in terms of other elements and relations, it is necessary that one or more of the elements and one or more of the relations between them remain entirely undefined; otherwise a vicious circle is unavoidable. Likewise certain of the propositions are regarded as fundamental, in the sense that all other propositions are derivable, as logical consequences, from these fundamental ones. But here again it is a logical necessity that one or more of the prop- ositions remain entirely improved ; otherwise a vicious circle is again inevitable. The starting point of any strictly logical treatment of geometry (and indeed of any "branch of mathematics) must then le a set of un- defined elements and relations, and a set of unproved propositions involving them ; and from these all other propositions (theorems) are to le derived ly the methods of formal logic. Moreover, since we assumed the point of view of formal (Le. symbolic) logic, the unde- fined elements are to be regarded as mere symbols devoid of content, except as implied by the fundamental propositions. Since it is mani- festly absurd to speak of a proposition involving these symbols as 1 2 INTEODUCTION [INTROD. self-evident, the unproved propositions referred to above must be re- garded as mere assumptions. It is customary to refer to these funda- mental propositions as axioms or postulates, but we prefer to retain the term assiwiption as more expressive 'of their real logical character. We understand the term a mathematical science to mean any set of propositions arranged according to a sequence of logical deduction. From the point of view developed above such a science is purely abstract. If any concrete system of things may be regarded as sat- isfying the fundamental assumptions, this system is a concrete ap- plication or representation of the abstract science. The practical importance or triviality of such a science depends simply on the importance or triviality of its possible applications. These ideas will be illustrated and further discussed in the next section, where it will appear that an abstract treatment has many advantages quite apart from that of logical rigor. 2. Consistency; categoricalness, independence. Example of a math- ematical science. The notion of a class * of objects is fundamental in logic and therefore in any mathematical science. The objects which make up the class are called the elements of the class. The notion of a class, moreover, and the relation of belonging to a class (being included in a class, being an element of a class, etc.) are primi- tive notions of logic, the meaning of which is not here called in question.f The developments of the preceding section may now be illustrated and other important conceptions introduced by considering a simple example of a mathematical science. To this end let S be a class, the elements of which we will denote by A 9 B,C,... Further, let there be certain undefined subclasses $ of S, any one of which we will call an m-class. Concerning the elements of S and the m-classes we now make the following ASSUMPTIONS : I. If A and J3 are distinct elements of S, there is at least one m-class containing both A and B. * Synonyms for class are set, aggregate, assemblage, totality; in German, Menge; inPrench, ensemble. 1 Of. B. Russell, The Principles of Mathematics, Cambridge, 1003 ; and L, Cou- turat, Les Principes des mathe'matiques, Paris, 1905. $ A class S' is said to be a subclass of another class S, if every element of S' is an element of S. 2] A MATHEMATICAL SCIENCE 3 II. If A and B are distinct elements of S, there is not more than one m-class containing loth A and B. III. Any two m-classes have at least one element of S in common. IV. There exists at least one m-class. ' V. JEJvery m-class contains at least three elements of S. VI. All the elements of S do not belong to the same m-class, VII. No m-class contains more than three elements of S. The reader will observe that in this set of assumptions we have just two undefined terms, viz., element of S and m-class, and one undefined relation, belonging to a class. The undefined terms, more- over, are entirely devoid of content except such as is implied in the assumptions. Now the first question to ask regarding a set of assumptions is : Are they logically consistent? In the example above, of a set of assumptions, the reader will find that the assumptions are all true statements, if the class S is interpreted to mean the digits 0, 1, 2, 3, 4, 5, 6 and the m-classes to mean the columns in the following table : 0123456 (1) 1234560 3456012 This interpretation is a concrete representation of our assumptions. Every proposition derived from the assumptions must be true of this system of triples. Hence none of the assumptions can be logically inconsistent with the rest ; otherwise contradictory statements would be true of this system of triples. Thus, in general, a set of assumptions is said to le consistent if a single concrete representation of the assumptions can le given* Knowing our assumptions to be consistent, we may proceed to de- rive some of the theorems of the mathematical science of which they are the basis : Any two distinct elements of S determine one and only one m-class containing "both these elements (Assumptions I, II). * It will be noted that this test for the consistency of a set of assumptions merely shifts the difficulty from one <do main to another. It is, however, at present the only test known. On the question as to the possibility of an absolute test of consistency, cf. Hilbert, Grundlagen der Geometrie, 2d ed., Leipzig (1903), p. 18, and Verhandlungen d. III. intern, math* Kongresses zu Heidelberg, Leipzig (1904), p. 174; Padoa, L'Enseignement niathe'matique, Vol. V (1903), p. 85. 4 INTEODUCTJOK [INTBOD. The m-class- containing the elements A and B may conveniently be denoted by the symbol AB. Any two m-classes have one and only one dement of S in common (Assumptions II, III). Tliere exist three elements of S which are not all in the same m-class (Assumptions IV, V, VI). In accordance with the last theorem, let A, B, C be three elements of S not in the same m-class. By Assumption V there must be a third element in each of the m-classes AB, BC, CA, and by Assump- tion II these elements must be distinct from each other and from A, By and C. Let the new elements be D, E, G, so that each of the triples ABD, BCE, CAG belongs to the same m-class. By Assumption III the m-classes AE and BG, which are distinct from aH the m-classes thus far obtained, have an element of S ia common, which, by Assumption II, is distinct from those hitherto mentioned ; let it be denoted by F, so that each of the triples AEF and BIG belong to the same m-class. No use has as yet been made of As- sumption VII. We have, then, the theorem : Any class S subject to Assumptions / VI contains at least seven elements. Now, making use of Assumption VII, we find that the m-classes thus far obtained contain only the elements mentioned. The m-classes CD and AEF have an element in common (by Assumption III) which cannot be A or E, and must therefore (by Assumption VII) be F. Similarly, AOG and the m-class DE have the element G in common. The seven elements A, B, C, I), E, F, G have now been arranged into m-classes according to the table A B C D E F G (!') B C D E F G A D E F G A B G in which the columns denote m-classes. The reader may note at once that this table is, except for the substitution of letters for digits, entirely equivalent to Table (1); indeed (I/) is obtained from (1) by replacing by A, 1 by B, 2 by (7, etc. We can show, furthermore, that S can contain no other elements than A, B, <7, D, E, F, G. For suppose there were another element, T. Then, by Assumption III, 2] CATEGORICALKESS 5 the m-classes TA and BFG would have an element in common. This element cannot be B> for then ABTD would belong to the same w-class ; it cannot be F, for then AFTJE would all belong to the same w-class; and it cannot be G, for then AGTC would all belong to the same m-class. These three possibilities all contradict Assumption VII. Hence the existence of T would imply the existence of four elements in the wi-class BFG, which is likewise contrary to Assumption VII. The properties of the class S and its m-classes may also be repre- sented vividly by the accompanying figure (fig. 1). Here we have represented the elements of S *by points (or spots) in a plane, and have joined by a line every triple of these points which form an m- class. It is seen that the points may be so chosen that all but one of these lines is a straight line. This suggests at once a similarity to ordinary plane geometry. Sup- pose we interpret the elements of S to be the points of a plane, and interpret the m-classes to be the straight lines of the plane, and let us reread our assumptions with this interpretation. Assumption VII is false, but all the others are true with the exception of Assumption III, which is also true except when the lines are parallel. How this exception can be removed we will discuss in the next section, so that we may also regard the ordinary plane geometry as a representation of Assumptions I- VI. Returning to our miniature mathematical science of triples, we are now in a position to answer another important question : To what eoo- tent do Assumptions I VII characterize the class S and the m-classes ? We have just seen that any class S satisfying these assumptions may be represented by Table (I/) merely by properly labeling the ele- ments of S. In other words, if S x and S 2 are two classes S subject to these assumptions, every element of S x may be made to correspond * to a unique element of S 2 , in such a way that every element of S 2 is the correspondent of a unique element of S y and that to every w-class of Sj there corresponds an m-class of S r The two classes are * The notion of correspondence is another primitive notion which we take over without discussion from the general logic of classes. 6 INTBODUCTION" [INTROB. then said to be in one-to-one reciprocal correspondence, or to be simply isomorphic* Two classes S are then abstractly equivalent ; i.e. there exists essentially only one class S satisfying Assumptions I- VII. This leads to the following fundamental notion : A set of assumptions is said to "be categorical, if there is essentially only one system for which the assumptions are 'valid ; i.e. if any, two such systems may le made simply isomorphic. We have just seen that the set of Assumptions I-VII is categor- ical. If, however, Assumption VII be omitted, the remaining set of six assumptions is not categorical. We have already observed the possibility of satisfying Assumptions I- VI by ordinary plane geom- try. Since Assumption III, however, occupies as yet a doubtful posi- tion in this interpretation, we give another, which, by virtue of its simplicity, is peculiarly adapted to make clear the distinction between categorical and noncategorical. The reader will find, namely, that each of the first six assumptions is satisfied by interpreting the class S to consist of the digits 0, 1, 2, - *, 12, arranged according to the fol- lowing table of m-classes, every column constituting one m-class : 0123456 789 10 11 12 9 1234567891011120 ^ 3 4 5 6 7 8 9 10 11 12 1 2 9 10 11 12 012 345 6 78 Henee Assumptions I-VI are not sufficient to characterize completely the class S, for it is evident that Systems (1) and (2) cannot be made isomorphic. On the other hand, it should be noted that all theorems derivable from Assumptions I-VI are valid for both (1) and (2). These two systems are two essentially different concrete representa- tions of the same mathematical science. This brings us to a third question regarding our assumptions : Are they independent ? That is, can any one of them be derived as a log- ical consequence of the others ? Table (2) is an example which shows that Assumption VII is independent of the others, because it shows that they can all be true of a system in which Assumption VII is false. Again, if the class S is taken-to mean the three letters A, B, (7, * The isomorphism of Systems (1) and (1") is clearly exhibited in fig, 1, where each point is labeled both with a digit and with a letter. This isomorphism may, moreover, be established in 7-64 different ways. 2,3] - IDEAL ELEMENTS 7 and the m-classes to consist of the pairs AB, JBC, CA, then it is clear that Assumptions I, II, III, IV, VI, VII are true of this class S, and therefore that any logical consequence of them is true with this interpretation. Assumption V, however, is false for this class, and cannot, therefore, be a logical consequence of the other assump- tions. In like manner, other examples can be constructed to show that each of the Assumptions I- VII is independent of the remain- ing ones. 3. Ideal elements in geometry. The miniature mathematical science which we have just been studying suggests what we must do on a larger scale in a geometry which describes our ordinary space. We must first choose a set of undefined elements and a set of funda- mental assumptions. This choice is in no way prescribed a priori, but, on the contrary, is very arbitrary. It is necessary only that the undefined symbols be such that all other elements and relations that occur are definable in terms of them ; and the fundamental assump- tions must satisfy the prime requirement of logical consistency, and be such that all other propositions are derivable from them by formal logic. It is desirable, further, that the assumptions be independent* and that certain sets of assumptions be categorical. There is, further, the desideratum of utmost symmetry and generality in the whole body of theorems. The latter means that the applicability of a theo- rem shall be as wide as possible. This has relation to the arrange- ment of the assumptions, and can be attained by using in the proof of each theorem a minimum of assumptions-! Symmetry can frequently be obtained by a judicious choice of terminology. This is well illustrated by the concept of "points at infinity" which is fundamental in any treatment of projective geome- try. Let us note first the reciprocal character of the relation expressed by the two statements : A point lies on a line. A line passes through a point. To exhibit clearly this reciprocal character, we agree to use the phrases A point is on a line ; A line is on a point '* This is obviously necessary for the precise distinction between an assumption and a theorem. t If the set of assumptions used in the proof of a theorem is not categorical, the applicability of the theorem is evidently -wider than in the contrary case. Cf . exam- ple of preceding section* g INTRODUCTION [INTROD. to express this relation. Let us now consider the following two propositions : 1. Any two distinct points of 1'. Any two distinct lines of a a plane are on one and only one plane are on one and only one Urn* P int - Either of these propositions is obtained from the other by simply interchanging the words point and line. The first of these propositions we recognize as true without exception in the ordinary Euclidean geometry. The second, however, has an exception when the two lines are parallel. In view of the symmetry of these two propositions it would clearly add much to the symmetry and generality of all propositions derivable from these two, if we could regard them both as true without exception. This can be accomplished by attributing to two parallel lines apoint of intersection. Such a point is not, of course, a point in the ordinary sense ; it is to be regarded as an ideal point, which we suppose two parallel lines to have in common. Its introduction amounts merely to a change in the ordinary termi- nology. Such an ideal point we call a point at infinity ; and we suppose one such point to exist on every line.f The use of this new term leads to a change in the statement, though not in the meaning, of many familiar propositions, and makes us modify the way in which we think of points, lines, etc. Two non- parallel lines cannot have in common a point at infinity without doing violence to propositions 1 and 1'; and since each of them has a point at infinity, there must be at least two such points. Proposition 1, then, requires that we attach a meaning to the notion of a line on two points at infinity. Such a line we call a line at infinity, and think of it as consisting of all the points at infinity in a plane. In like manner, if we do not confine ourselves to the points of a single plane, it is found desirable to introduce the notion of a plane through three points at infinity which are not all on the same line at infinity. Such a plane we call a plane at infinity, and we think * By line throughout we mean straight line. f It should be noted that (since we are taking the point of view of Euclid) we do not think of a line as containing more than one point at infinity ; for the supposi- tion that a line contains two such points would imply either that two parallels can be drawn through a given point to a given line, or that two distinct lines can have more than one point in common. 3,4] CONSISTENCY OF IDEAL ELEMENTS 9 of it as consisting of all the points at infinity in space. Every ordi- nary plane is supposed to contain just one line at infinity ; every sys- tem of parallel planes in space is supposed to have a line at infinity in common with the plane at infinity, etc. The fact that we have difficulty in presenting to our imagination the notions of a point at infinity on a line, the line at infinity in a plane, and the plane at infinity in space, need not disturb us in this connection, provided we can satisfy ourselves that the new terminol- ogy is self-consistent and cannot lead to contradictions. The latter condition amounts, in the treatment that follows, simply to the con- dition that the assumptions on which we build the subsequent theory be consistent. That they are consistent will be shown at the time they are introduced. The use of the new terminology may, however, be justified on the basis of ordinary analytic geometry. This we do in the next section, the developments of which will, moreover, be used frequently in the sequel for proving the consistency of the assumptions there made. 4. Consistency of the notion of points, lines, and plane at infinity. We will now reduce the question of the consistency of our new ter- minology to that of the consistency of an algebraic system. For this purpose we presuppose a knowledge of the elements of analytic geom- etry of three dimensions.* In this geometry a point is equivalent to a set of three numbers (x, y, z). The totality of all such sets of numbers constitute the analytic space of three dimensions. If the 'numbers are all real numbers, we are dealing with the ordinary "real" space ; if they are any complex numbers, we are dealing with the ordi- nary " complex " space of three dimensions. The following discussion applies to either case. A plane is the set of all points (number triads) which satisfy a single linear equation ax + ly + cz + d = 0. A Ime is the set of all points which satisfy two linear equations, a^x + \y + CjZ + d^ 0, 0, * Such knowledge is not presupposed elsewhere in this hook, except in the case of consistency proofs. The elements of analytic geometry are indeed developed from the beginning (cf. Chaps. VI, VH). 10 INTRODUCTION [INTEOD. provided the relations do not hold.* Now the points (x, y, z), with the exception of (0, 0, 0), may also be denoted by the direction cosines of the line joining the point to the origin of coordinates and the distance of the point from the origin ; say by f I, m,n, -)> dj where d = V^+^+r, and I = -> m = --> n = ^ The origin itself d d cL may be denoted by (0, 0, 0, k), where i is arbitrary. Moreover, any four numbers (x v aj a , 8 , a 4 ) (x = 0), proportional respectively to I, m, n, - )> will serve equally well to represent the point (x, y, ), provided we agree that (x v x 2 , x s , x 4 ) and (cx v cx^ c% s , cx 4 ) represent the same point for all values of c different from 0. For a point (x 9 y, z) determines ex 7 c?/ J cmi, where e is arbitrary (c = 0), and (x v x v x v a; 4 ) determines /-t \ ^1 ^o ^o (1) # = -^y=:->;s:=-3> aj 4 a? 4 as, provided a? 4 = 0. We have not assigned a meaning to (x v # 2 , x s , x 4 ) when ^ 4 = 0, but it is evident that if the point ( d, cm, en, - ) moves away from the \ *v origin an unlimited distance on the line whose direction cosines are /, m, n, its coordinates approach (cl, cm, en, 0). A little consideration will show that as a point moves on any other line with direction * It should be noted that we are not yet, in this section, supposing anything known regarding points, lines, etc., at infinity, but are placing ourselves on the "basis of elementary geometry. 4] CONSISTENCY OF IDEAL ELEMENTS 11 cosines Z, m, n, so that its distance from the origin increases indefi- nitely, its coordinates also approach (cl, cm, en, 0). Furthermore, these values are approached, no matter in which of the two opposite direc- tions the point moves away from the origin. We now define (x v # 3 , # 8 , 0) as a point at infinity or an ideal point. We have thus associ- ated with every set of four numbers (x v x z , x s , x^ a point, ordinary or ideal, with the exception of the set (0, 0, 0, 0), which we exclude entirely from the discussion. The ordinary points are those for which x is not zero, and its ordinary Cartesian coordinates are given by the equations (1). The ideal points are those for which # 4 = 0. The num- bers (x v # 2 , # 3 , x) we call the homogeneous coordinates of the point. We now define a plane to be the set of all points (x v # 2 , # 8 , # 4 ) which satisfy a linear homogeneous equation : ax^+ bx% -f cx 9 + dx^ = 0. It is at once clear from the preceding discussion that as far as all ordinary points are concerned, this definition is equivalent to the one given at the beginning of this section. However, according to this definition all the ideal points constitute a plane # 4 = 0. This plane we call the plane at infinity. In like manner, we define a line to consist of all points (x v x 2 , X B> 4 ) which satisfy two distinct linear homogeneous equations : a^+ \x 2 + c&+ d&= 0, cx + d = 0. Since these expressions are to be distinct, the corresponding coefficients throughout must not be proportional. According to this definition the points common to any plane (not the plane at infinity) and the plane # 4 = constitute a line. Such a line we call a line at infinity, and there is one such in every ordinary plane. Finally, the line de- fined above by two equations contains one and only one point with coordinates (x v x v x s , 0) ; that is, an ordinary line contains one and only one point at infinity. It is readily seen, moreover, that with the above definitions two parallel lines have their points at infinity in common. Our discussion has now led us to an analytic definition of what may be called, for the present, an analytic projectile space of three dimensions. It consists of : 12 INTRODUCTION [INTROD. Points : All sets of four numbers (x v x# X B) # 4 ), except the set (0, 0, 0, 0), where (cx v cx^ cx z) cxj is regarded as identical with (x# # 2 , s g , # 4 ), provided c is not zero. Planes: All sets of points satisfying one linear homogeneous equation. lines: All sets of points satisfying two distinct linear homoge- neous equations. Such a protective space cannot involve contradictions unless our ordinary system of real or complex algebra is inconsistent. The defi- nitions here made of points, lines, and the plane at infinity are, however, precisely equivalent to the corresponding notions of the preceding section. We may therefore use these notions precisely in the same way that we consider ordinary points, lines, and planes. Indeed, the fact that no exceptional properties attach to our ideal elements follows at once from the symmetry of the analytic formu- lation; the coordinate # 4 , whose vanishing gives rise to the ideal points, occupies no exceptional position in the algebra of the homo- geneous equations. The ideal points, then, are not to be regarded as different from the ordinary points. All the assumptions we shall make in our treatment of projective geometry will be found to be satisfied by the above analytic creation, which therefore constitutes a proof of the consistency of the assump- tions in question. This the reader will verify later. 5. Projective and metric geometry. In projective geometry no distinction is made between ordinary points and points at infinity, and it is evident by a reference forward that our assumptions pro- vide for no such distinction. We proceed to explain this a little more fully, and will at the same time indicate in a general way the difference between projective and the ordinary Euclidean metric geometry. Confining ourselves first to the plane, let m and m r be two distinct lines, and P a point not on either of the two lines. Then the points of m may be made to correspond to the points of m/ as follows : To every point A on m let correspond that point A r on m f in which m f meets the line joining A to P (fig. 2). In this way every point on either line is assigned a unique corresponding point on the other line. This type of correspondence is called perspective, and the points on one line are said to be transformed into the points of the other by 4, 5] PEOJECTIYE AND METRIC GEOMETBY 13 a perspective transformation with center P. If the points of a line m be transformed into the points of a line m 1 by a perspective transfor- mation with center P, and then the points of m 1 be transformed into the points of a third line m ff by a perspective transformation with a new center Q ; and if this be continued any finite number of times, ulti- mately the points of the line m will have been brought into corre- spondence with the points of a line m (n) , say, in such a way that every point of m corresponds to a unique point of m (n \ A correspondence obtained in this way is called projectile, and the points of m are said to have been transformed into the points of m (n) by a protective transformation. Similarly, in three-dimensional space, if lines are drawn joining every point of a plane figure to a fixed point P not in the plane TT of the figure, then the points in which this totality of lines meets another plane 7r f will form a new figure, such that to every point of TT will correspond a unique point of TT', and to every line of TT will correspond a unique line of TT'. We say that the figure in TT has been transformed into the figure in TT' by & perspective transformation with center P. If a plane figure be subjected to a succession of such per- spective transformations with different centers, the final figure will still be such that its points and lines correspond uniquely to the points and lines of the original figure. Such a transformation is again called a projectile transformation. In projective geometry two figures that may be made to correspond to each other by means of a projec- tive transformation are not regarded as different. In other words, 14 INTRODUCTION [INTROD. protective geometry is concerned with those properties of figures that are left unchanged when the figures are subjected to a projective transformation. It is evident that no properties that involve essentially the notion of measurement can have any place in projective geometry as such ;* hence the term projective, to distinguish it from the ordinary geom- etry, which is almost exclusively concerned with properties involving the idea of measurement. In case of a plane figure, a perspective transformation is clearly equivalent to the change brought about in the aspect of a figure by looking at it from a different angle, the observer's eye being the center of the perspective transformation. The properties of the aspect of a figure that remain unaltered when the observer changes his position will then be properties with which projective geometry concerns itself. For this reason von Staudt called this science Geometric der Lage. In regard to the points and lines at infinity, we can now see why they cannot be treated as in any way different from the ordinary points and lines of a figure. For, in the example given of a per- spective transformation between lines, it is clear that to the point at infinity on m corresponds in general an ordinary point on m ! 9 and conversely. And in the example given of a perspective transforma- tion between planes we see that to the line at infinity in one plane corresponds in general an ordinary line in the other. In projective geometry, then, there can be no distinction between the ordinary and the ideal elements of space. * The theorems of metric geometry may however be regarded as special cases of projective theorems. CHAPTER I THEOREMS OF ALIGNMENT AND THE PRINCIPLE OF DUALITY 6. The assumptions of alignment. In the following treatment of protective geometry we have chosen the point and the line as unde- fined elements. We consider a class (cf. 2, p. 2) the elements of which we call points, and certain undefined classes of points which we call lines. Here the words point and line are to be regarded as mere symbols devoid of all content except as implied in the as- sumptions (presently to be made) concerning them, and which may represent any elements for which the latter may le valid propositions. In other words, these elements are not to be considered as having properties in common with the points and lines of ordinary Euclidean geometry, except in so far as such properties are formal logical conse- quences of explicitly stated assumptions. We shall in the future generally use the capital letters of the alphabet, as A, B, C, P, etc., as names for points, and the small let- ters, as a, &, c, Z, etc., as names for lines. If A and B denote the same point, this will be expressed by the relation A = B ; if they repre- sent distinct points, by the relation A^B. If A = B, it is sometimes said that A coincides with B, or that A is coincident with B. The same remarks apply to two lines, or indeed to any two elements of the same kind. All the relations used are defined in general logical terms, mainly by means of the relation of 'belonging to a class and the notion of one- to-one correspondence. In case a point is an element of one of the classes of points which we call lines, we shall express this relation by any one of the phrases : the point is on or lies on or is a point of the line, or is united with the line ; the line passes through or con- tains or is united with the point. We shall often find it convenient to use also the phrase the line is on the point to express this relation. Indeed, all the assumptions and theorems in this chapter will be stated consistently in this way. The reader will quickly become ac- customed to this " on " language, which is introduced with the purpose 15 16 THEOREMS OF ALIGNMENT AND DUALITY [CHAJP.I of exhibiting in its most elegant form one of the most far-reaching theorems of projective geometry (Theorem 11). Two lines which have a point in common are said to intersect in or to meet in that point, or to le on a common point. Also, if two distinct points lie on the same line, the line is said to join the points. Points which are on the same line are said to be collinear ; points which are not on the same line are said to be noncollinear. Lines which are on the same point (i.e. contain the same point) are said to be copunctal, or concurrent* Concerning points and lines we now make the following assump- tions : THE ASSUMPTIONS OF ALIGNMENT, A : A 1. If A and B are distinct points, there is at least one line on both A and B. A 2. If A and B are distinct points, there is not more than one line on both A and B. A3. If A, B, are points not all on the same line, and D and E (D = E] are points such that B, C, D are on a line and C, A, E are on a line, there is a point F such that A, B, F are on a line and also D, E, F arc on a line (fig. 3).f It should be noted that this set of assumptions is satisfied by the triple system (1), p. 3, and also by the system of quadruples (2), p. 6, as well as by the points and lines of ordinary Euclidean geom- etry with the notion of " points at infinity " (cf. 3, p. 8), and by * The object of this paragraph is simply to define the terms in. common use in terms of the general logical notion of belonging to a class. In later portions of this book we may omit the explicit definition of such common terms when such definition is obvious. f The figures are to be regarded as a concrete representation of our science, in which the undefined " points" and "lines" of the science are represented by points and lines of ordinary Euclidean geometry (this requires the notion of ideal points; cf. 3, p. 8). Their function is not merely to exhibit one of the many possible concrete representations, but also to help keep in mind the various rela- tions in question. In using them, however, great care must be exercised not to use any properties of such figures that are not formal logical consequences of the assumptions ; in other words, care must be taken that all deductions are made formally from the assumptions and theorems previously derived from the assumptions* * 7] THE PLANE 17 the " analytic projective space " described in 4. Any one of these representations shows that our set of Assumptions A is consistent* The following three theorems are immediate consequences of the first two assumptions. THEOREM 1. Two distinct points are on one and only one line. (Al,A2)f The line determined by the points A, B (A = B) will often be denoted by the symbol or name AB. THEOREM 2. If C and D (C = D) are points on the line AB, A and B are points on the line CD. (Al, A 2) THEOREM 3. Two distinct lines cannot be on more than one common point. (A 1, A 2) Assumption A3 will be used in the derivation of the next theo- rem. It may be noted that under Assumptions A 1, A 2 it may be stated more conveniently as follows : If A 9 B, C^are ^points not jtll^on the samej-ine, the line joining any point -Z> on the. line BO to _any point JEJ (D = E) on the line CA meets the line AB in a point F. This is the form in which this assumption" is~ generally "used in the sequel. 7. The plane. DEFINITION. If P t Q, E are three points not on the same line, and I is a line joining Q and R, the class S 2 of all points such that every point of S 2 is collinear with P and some point of I is called the plane determined by P and I. We shall use the small letters of the Greek alphabet, a, /3, y, TT, etc., as names for planes. It follows at once from the definition that P and every point of I are points of the plane determined by P and I. THEOREM 4. If A and B are points on a plane TT, then every point on the line AB is on TT. (A) Proof. Let the plane TT under consideration be determined by the point P and the line I * In the multiplicity of the possible concrete representations is seen one of the great advantages of the formal treatment quite aside from that of logical rigor. It is clear that there is a great gain in generality as long as the fundamental assump- tions are not categorical (cf. p. 6). In the present treatment our assumptions are not made categorical until yery late. t The symbols placed in parentheses after a theorem indicate the assumptions needed in its proof. The symbol A will be used to denote the whole set of Assump- tions A 1, A 2, A3. FIG. 18 THEOREMS OF ALIGNMENT AND DUALITY [CHAP. I 1. If both A and B are on Z, or if the line AB contains P, the theorem is immediate. 2. Suppose A is on I, B not on I, and AB does not contain P (fig. 4). Since B is a point of w, there is a point 2?' on I collinear with B and P. If (7 be any point on AB 9 the line joining C on -4J5 to P on .B1?' will have a point T in common with AB'=l (A 3). Hence is a point of TT. 3. Suppose neither A nor .2? is on I and that -4J? does not con- tain P (fig. 5). Since A and .# are points of TT, there exist two points J 7 and ff on Z collinear with A, P and 5, P respectively. The line join- ing A on A'P to .# on PB 1 has a point Q in common with J?'.4' (A 3), Hence every point of the line AB AQ is a point of TT, by the preceding case. This completes the proof. If all the points of a line are points of a plane, the line is said to be a line of the plane, or to lie in or to le in or to le on the plane; the plane is said to pass through, or to contain the line, or we may also say the plane is on the line. Further, a point of a plane is said to le in or to lie in the plane, and the plane is on the point 8. The first assumption of extension. The theorems of the pre- ceding section were stated and proved on the assumption (explicitly stated in each case) that the necessary points and lines exist. The assumptions of extension, M, insuring the existence of all the points which we consider, will be given presently. The first of these, how- ever, it is desirable to introduce at this point. AN ASSUMPTION OF EXTENSION : E 0. There are at least three points on every line. This assumption is needed in the proof of the following THEOREM 5. Any two lines on the same plane TT are on a common point. (A, E 0) I FIG. 5 8] ASSUMPTION OF EXTENSION 19 Proof. Let the plane TT be determined by the point P and the line I, and let a and 6 be two distinct lines of TT. 1. Suppose a coincides with I (fig. 6). If I contains P, any point B of 6 (E 0) is colHnear with P and some point of Z=a, which proves the theorem when I contains P. If I does not contain P } there exist on 5 two points A and I? not on I (E 0), and since they are points of TT, they are collinear with P and two points A 1 and B ! of / respectively. The line joining A on A'P to .# on PB 1 has a point JS in common with A'B 1 (A 3) ine^ theplane PIG-. 6 i.e. I = a and & have a point in common. Hence TT has a point in common with L 2. Let a and 5 both be distinct from L (i) Let a contain P (fig. 7). The line joining P to any point B of 5 (E 0) has a point B r in com- mon with I (Case 1 of this proof). Also the lines a and 5 have points ,4' and R respectively in common with I (Case 1). Now the line A!P = a contains the points A J of EB } and P of ^^ and hence has a point A in common with JFia. 7 Hence in common = Z>. h as a point it]^ any line om gA P. (ii) Let neither a nor contain P (fig. 8). As before, & and 6 meet Z in two points Q and 2 respectively. Let B 1 be a point of I distinct from Q and E (E 0). The line PB r then meets a and 6 in two points A and .Z? respectively (Case 2, (i)). If J[ = B y the theorem is proved. If A = .#, the line 6 has the point E in common with Q^ r and the point B in common with B f A, and hence has a point in common with AQ = a (A3). 20 THEOREMS OF ALIGNMENT AND DUALITY [CHAP, i THEOREM 6. Tlie plane a determined ly a line I and a point P is identical with the plane /3 determined ly a line m and a point Q, provided m and Q are on a. (A, E 0) Proof. Any point B of /3 is collinear with Q and a point A of m (fig. 9). A and Q are both points of a, and hence every point of the line AQ is a point of a (Theorem 4). Hence every point of /3 is a point of a. Conversely, let B be any point of a. The line BQ meets w in a point (Theorem 5). Hence every point of a is also a point of /J. COROLLARY. There is one and only one plane determined "by three non- collinear points, or ly a line and a point not on the line, or ly two inter- secting lines. (A, E 0) The data of the corollary are all equivalent by virtue of EO. We will denote by ABC the plane determined by the points A, B, C\ by aA the plane determined by the line a and the point A 9 etc. THEOREM 7. Two distinct planes which are on two common points A y B (A = 3) are on all the points of the line AB, and on no other com- , mon points. (A, EO) Proof. By Theorem 4 the line AS lies in each of the two planes, which proves the first part of the proposition. Suppose C 9 not on AB, were a point common to the two planes. Then the plane determined by ,A, B, C would be identical with each of the given planes (Theorem 6), which contradicts the hypothesis that the planes are distinct. COROLLARY. Two distinct planes cannot "be on more than one com- mon line. (A, E 0) 9. jnbiejhjee-space. DEFINITION. If P, Q, E, T are four points not in the same plane, and if TT is a plane containing Q, R, and T, the class S g of all points such that every point of S 3 is- collinear with P and some point of TT is called the space of three dimensions , or the three-space determined by P and TT. If a point belongs to a three-space or is a point of a three-space, it is said to le in or to lie in or to le on the three-space. If all the points of a line or plane are points of a three-space S 3 , the line or plane is said 9] THE THREE-SPACE 21 FHK 10 to lie in or to le in or to le on the S 3 . Also the three-space is said to le on the point, line, or plane. It is clear from the definition that P and every point of TT are points of the three-space determined by P and TT. THEOREM 8. If A and B are distinct points on a three-space S s , every point on the line AB is on S 3 . (A) Proof. Let S 3 be determined by a plane TT and a point P. 1. If A and B are both in TT, the theorem is an immediate conse- quence of Theorem 4. 2. If the line AB contains P, the theorem is obvious. 3. Suppose A is in TT, B not in TT ; and AB does not contain P (fig. 10). There then exists a point S f (= A) of TT collinear with B and P (def.). The line joining any point M on AB to P on BB f has a point M f in common with 2^-4 (A 3). But M f is a point of w, since it is a point of AB f . Hence jYis a point of S 3 (del). 4. Let neither A nor J5 lie in TT, and let AB not contain P (fig. 11). The lines PA and P# meet TT in two points A r and B r respectively. But the line joining A on A r P to B on P-B' has a point C in common with B r A r . C is a point of TT, which reduces the proof to Case 3. It may be noted that in this proof no use has been made of E 0. In discussing Case 4 we have proved incidentally, in connection with EO and 'Theorem 4, the fol- lowing corollary: COROLLARY 1. If S 3 is a three-space determined ly a point P and a plane TT, then TT and any line on S 3 but not on TT are on one and only one common point (A, E 0) COROLLARY 2. JEvery point on any plane determined by three non- eollinear points on a three-space S 8 is on S s . (A) FIG. 11 22 THEOREMS OF ALIGNMENT AND DUALITY [CHAP. I Proof. As before, let the three-space be determined by TT and P, and let the three noncollinear points 'be A, B, 0. Every point of the line BC is a point of S s (Theorem 8), and every point of the plane ABC* is collinear with A and some point of BC. COROLLARY 3. If a three-space S 8 is determined ly a point P and a plane TT, then TT and any plane on S 3 distinct from TT are on one and only one common line. (A, E 0) Proof. Any plane contains at least three lines not passing through the same point (del, A 1). Two of these lines must meet TT in two distinct points, which are also points of the plane of the lines (Cor. 1). The result then follows from Theorem 7. THEOREM 9. If a plane a and a line a not on a are on the same threes-space S gJ then a and a are on one and only one common point. (A,EO) Proof. Let S 3 be determined by the plane TT and the point P. 1. If a coincides wither, the theo- rem reduces to Cor. 1 of Theorems. 2. If # is distinct from TT, it has a line I in common with TT (Theorem 8, Cor. 3). Let A be any point on a not on a (EO) (fig. 12), The plane aA, determined by A and a, meets TT in a line m & I (Theorem 8, Cor. 3). The lines I, m have a point B in common (Theorem 5). The line AB in aA meets a in a point Q (Theorem 5), which is on a, since AB is on a. That a and a have no other point in common follows from Theorem 4. COROLLARY 1. Any two distinct planes on a three-space are on one and only one common line. (A, E 0) The proof is similar to that of Theorem 8, Cor. 3, and is left as an exercise. COROLLARY 2. Conversely, if two planes are on a common line, there exists a three-space on loth. (A, E 0) * The proof can evidently be so worded as not to imply Theorem 6. FIG. 12 9] THE THREE-SPACE 23 Proof. If the planes a and ft are distinct and have a line I in common, any point P of /3 not on I "will determine with a a three- space containing I and P and hence containing ft (Theorem 8, Cor. 2). COROLLARY 3. Three planes on a three-space ivhich are not on a common line are on one and only one common point. (A, E 0) Proof. This follows without difficult} 7 * from the theorem and Cor. 1. Two planes are said to determine the line which they have in com- mon, and to intersect or meet in that line. Likewise if three planes have a point in common, they are said to intersect or meet in the point. COROLLARY 4. If a, y3, 7 are three distinct planes on the same S 3 but not on the same line, and if a line I is on each of t'wo planes p, v which are on the lines /3<y and ya respectively, then it is on a plane \ which is on the line aft. (A, E 0) p , r Q Proof. By Cor. 3 the planes a, I ,r j3, 7 have a point P in common, /\| 7 ~~7 7T7 so that the lines /3y, ya, a/3 all XT ^-9 -f- -i contain P. The line I, being com- / X v / s x / / mon to planes through @y and ya, must pass through P, and the lines I and a/3 therefore intersect in P and hence determine a plane \ (Theorem 6, Cor.). THEOREM 10. The three-space S 3 determined by a plane TT, and a point P is identical with the three-space S determined 1y a plane 7r f and a point P', provided TT' and P f are on S 3 . (A, EO) Proof. Any point A of S^ (fig. 13) is collinear with P 1 and some point A f of TT'; but P f and A f are both points of S 3 and hence A is a point of S 3 (Theorem 8). Hence every point of S^ is a point of S s . Conversely, if A is any point of S 3 , the line AP f meets TT' in a point (Theorem 9). Hence every point of S s is also a point of S. COROLLARY. TJiere is one and only one three-space on four given points not on the same plane, or a plane and a line not on the plane, or two nonintersecting lines. (A, E 0) The last part of the corollary follows from the fact that two nonintersecting lines are equivalent to four points not in the same plane (EO). 24 THEOREMS OF ALIGNMENT AND DUALITY [CHAP. I It is convenient to use tlie term coplanar to describe points in the same plane. And we shall use the term skew lines for lines that have no point in common. Four noncoplanar points or two skew lines are said to determine the three-space in which they lie. 10. The remaining assumptions of extension for a space of three dimensions. In 8 we gave a first assumption of extension. We will now add the assumptions which insure the existence of a space of three dimensions, and will exclude from our consideration spaces of higher dimensionality. ASSUMPTIONS OF EXTENSION, E : E.l. There exists at least one line. E 2. All points are not on the same line. E 3. All points are not on the same plane. E 3'. If S 3 is a three-space, every point is on S 3 . The last may be called an assumption of closure.* The last assumption might be replaced by any one of several equiv- alent propositions, such as for example : Every set of five points lie on the same three-space ; or Any two distinct planes have a line in common. (Of. Cor. 2, Theo- rem 9) There is no logical difficulty, moreover, in replacing the assumption (E3 ; ) of closure given above by an assumption that all the points are not on the same three-space, and then to define a " four-space " in a manner entirely analogous to the definitions of the plane and to the three-space already given. And indeed a meaning can be given to the words point and line such that this last assumption is satisfied as well as those that precede it (excepting E3 ; of course). We could thus proceed step by step to define the notion of a linear space of any number of dimensions and derive the fundamental properties of alignment for such a space. But that is aside from our present purpose. The derivation of these properties for a four-space will furnish an excellent exercise, however, in the formal reasoning here emphasized (cf. Ex. 4, p. 25). The treatment for the ^dimensional case will be found in 12, p. 29. * The terms extension and closure in this connection were suggested by N. J. Lennes. It will be observed that the notation has been so chosen that Ei insures the exist- ence of a space of i dimensions, the line and the plane, being regarded as spaces of one and two dimensions respectively. 10] ASSUMPTIONS OF EXTENSION 25 The following corollaries of extension are readily derived from the assumptions just made. The proofs are left as exercises. COROLLARY 1. At least three coplanar lines are on every point. COROLLARY 2. At least three distinct planes are on every line. COROLLARY 3. All planes are not on the same line. COROLLARY 4. All planes are not on the same point. COROLLARY 5. If $ 3 is a three-space, every plane is on $ 8 . EXERCISES 1. Prove that through a given point P not on either of two skew lines I and I' there is one and only one line meeting both the lines Z, V. 2. Prove that any two lines, each of which meets three given skew lines, are skew to each other. _^ 3. Our assumptions do not as yet determine whether the number of points on a line is finite or infinite. Assuming that the number of points on one line is finite and equal to n + 1, prove that i. the number of points on every line is n + 1; ii. the number of points on every plane is n 2 + n + 1; iii. the number of points on every three-space is n 8 + n 2 + n + 1; *& iv. the number of lines on a three-space is (n 2 + 1) (n z + n + 1); v. the number of lines meeting any two skew lines on a three-space is *< vi. the number of lines on a point or on a plane is n 2 + n + 1. - -* 4. Using the definition below, prove the following theorems of alignment for a four-space on the basis of Assumptions A and E : DEFINITION. If P, Q, R, S, T are five points not on the same three-space, and S 3 is a three-space on Q, R, S, T, the class S 4 of a.11 points such that every point of S 4 is collinear with P and some point of S 3 is called the four-space determined by P and S 8 . i. If A and B are distinct points on a four-space, every point on the line A B is on the four-space. \ ii. Every line on a four-space which is not on a given three-space of the four-space has one and only one point in common with the three-space. iii. Every point on any plane determined by three noncollinear points on a four-space is on the four-space. iv. Every point on a three-space determined by four noncoplanar points of a four-space is on the four-space. v. Every plane of a four-space determined by a point P and a three-space S 3 has one and only one Jjne in common with S 3 , provided the plane is not on S 3 . vi. Every three-space on a four-space determined by a point P and a three- space S 3 has one and only one plane in common with S 3 , provided it does not coincide with S a . 26 THEOREMS OF ALIGNMENT AND DUALITY [CHAP. I viL If a three-space S 3 and a plane a not on S 3 are on the same four-space, S 3 and a have one and only one line in common. viii. If a three-space S 3 and a line I not on S g are on the same four-space, S 3 and I have one and only one point in common. ix. Two planes on the same four-space hut not on the same three-space have one and only one point in common. x. Any two distinct three-spaces on the same four-space have one and only one plane in common. xi. If two three-spaces have a plane in common, they lie in the same four-space. xii. The four-space S 4 determined l>y a three-space S 3 and a point P is identical with the four-space determined by a three-space 83 and a point P', provided 83 and P' are on S 4 . 5. On the assumption that a line contains n + 1 points, extend the results of Ex. 3 to a four-space. 11. The principle of duality. It is in order to exhibit the theorem of duality as clearly as possible that we have introduced the syrn- metrical, if not always elegant, terminology : A point is on a line. A line is on a point. A point is on a plane. A plane is on a point. A line is on a plane. A plane is on a line. A point is on a three-space. A three-space is on a point. A line is on a three-space. A three-space is on a line. A plane is on a three-space. A three-space is on a plane. The theorem in question rests on the f olio wing observation : If any one of the preceding assumptions, theorems, or corollaries is expressed by means of this "on" terminology and then a new proposition is formed by simply interchanging the words point and plane, then this new proposition will be valid, i.e. will be a logical consequence of the Assumptions A and E. We give below, on the left, a complete list of the assumptions thus far made, expressed in the " on " termi- nology, and have placed on the right, opposite each, the corresponding proposition obtained by interchanging the words point and plane together with the reference to the place where the latter proposition occurs in the preceding sections : ASSUMPTIONS A 1, A 2. If u4and THEOREM 9, COR. 1. If a and /9 are distinct points, there is one are distinct planes, there is one and and only one line on A and B. only one line on a and /3.* * By virtue of Assumption E 8' it is not necessary to impose the condition that the elements to be considered are in the same three-space. This observation should empha- size, however, that the assumption of closure is essential in the theorem to he nrovad ill] THE PKINCIPLE OF DUALITY 27 ASSUMPTION A 3. If A, B, Q are points not all on the same line, and D and JS (D 3* E] are points such that B, C 9 D are on a line and 9 A, E are 011 a line, then there is a point F such that A, B, F are on a line and also D, fi, F are on a line. ASSUMPTION EO. There are at least three points on every line. ASSUMPTION El. There exists at least one line. ASSUMPTION E 2. All points are not on the same line. ASSUMPTION E 3. All points are not on the same plane. ASSUMPTION E3'. If S, is a o three-space, every point is on S 3 . THEOBEM 9, COR. 4. If a, ft, <y are planes not all on the same line, and ^ and v(p*pv) are planes such that , % //, are on a line and % a, v are on a line, then there is a plane X such that a, /3, X are on a line and also /, v, X are on a line. COR. 2, p. 25. There are at least three planes on every line. ASSUMPTION E 1. , There exists at least one line. COR. 3, p. 25. All planes are not on the same line. COB. 4, P- 25. All planes are not on the same point. COE. 5, p. 25. If S s is a three- space, every plane is on S s . In all these propositions it is to be noted that a line is a class of points whose properties are determined by the assumptions, while a plane is a class of points specified by a definition. This definition in the "on" language is given below on the left, together with a* definition obtained from it by the interchange of point and plane. Two statements in this relation to one another are referred to as (space) duals of one another. If P, Q, JR are points not on the same line, and I is a line on Q and It, the class S 2 of all points such that every point of S 2 is on a line with P and some If X, JJL, v are planes not on the same line, and I is a line on" ^ and v > the class B 2 of all planes such that every plane of B 2 is on a line with X and some plane on I is called the "bundle determined by X and I. point on I is called the plane determined by P and I Now it is evident that, since X, /A, v and I all pass through a point 0> the bundle determined by X and I is simply the class of all planes on- the point 0. In like manner, it is evident that the dual of the defini- tion of a three-space is simply a definition of the class of all planes on a three-space. Moreover, dual to the class of all planes on a line we have the class of all points on a line, ie. the line itself, and conversely. 28 THEOBEMS OF ALIGNMENT AND DUALITY [CHAP. I With the aid of these observations we are now ready to establish the so-called principle of duality : THEOEEM 11. THE J^J^E^OF^DUALITY^FOR A SPACE OF THREE DIMENSIONS. Any proposition deducible from Assumptions A and E concerning points, lines, and planes of a three-space remains valid, if stated in the "ow-^Jcraj^fo^j^A^ the words "point" and "plane^ are intercJianqed. (A, E) Proof. Any proposition deducible from Assumptions A and E is obtained from the assumptions given above on the left by a certain sequence of formal logical inferences. Clearly the same sequence of logical inferences may be applied to the corresponding propositions given above on the right. They will, of course, refer to the class of all planes on a line when the original argument refers to the class of all points on a line, i.e. to a line, and to a bundle of planes when the original argument refers to a plane. The steps of the original argu- ment lead to a conclusion necessarily stated in terms of some or all of the twelve types of " on " statements enumerated at the beginning of this section. The derived argument leads in the same way to a conclusion which, whenever the original states that a point P is on a line l y says that a plane TT' is one of the class of planes on a line V, i.e. that TT' is on V\ or which, whenever the original argument states that a plane TT is on a point P, says that a bundle of planes on a point P f contains a plane -TT', Le. that P r is on TT'. Applying similar considerations to each of the twelve types of " on " statements in succession, we see that to each statement in the conclusion arrived at by the original argument corresponds a statement arrived at by the derived argument in which the words point and plane in the original statement have been simply interchanged. Any proposition obtained in accordance with the principle of dual- ity just proved is called the space dual of the original proposition. The point and plane are said to be dual elements ; the line is self- duaL We may derive from the above similar theorems on duality ^in a plane and at a point. For, consider a plane TT and a point P not on TT, together with all the lines joining P with every point of TT. Then to every point of TT will correspond a line through P, and to every line of TT will correspond a plane through P. Hence every proposi- tion concerning the points and lines of TT is also valid for the corre- sponding lines and planes through P. The space dual of the latter ii, 12] SPACE OF N DIMENSIONS 29 proposition is a new proposition concerning lines and points on a plane, which could have been obtained directly by interchanging the words point and line in the original proposition, supposing the latter to be expressed in the "on" language. This gives THEOREM 12. THE THEOREM OF DUALITY IN A PLANE. Any prop- osition deducible from Assumptions A and E concerning the points and lines of a plan&remains valid, if stated in the "on" terminology, when the words "point" and "line" are interchanged. (A, E) The space dual of this theorem then gives THEOREM 13. THE THEOREM OF DUALITY AT A POINT. Any prop- osition deducible from Assumptions A and E concerning the planes^ ^alid y if stated in the "on" termi- nology, when the words "plane" and " line" are interchanged. (A, E) The principle of duality was first stated explicitly by Gergonne (1826), but was led up to by the writings of Poncelet and others during the first quarter of the nineteenth century. It should be noted that this principle was for several years after its publication the subject of much discussion and often acrimonious dispute, and the treatment of this principle in many standard texts is far from convincing. The method of formal inference from explicitly stated assumptions makes the theorems appear almost self-evident. This may well be regarded as one of the important advantages of this method. It is highly desirable that the reader gain proficiency in forming the duals of given propositions. It is therefore suggested as an exercise that he state the duals of each of the theorems and corollaries in this chapter. He should in this case state both the original and the dual proposition in the ordinary terminology in order to gain facility in dualizing propositions without first stating them in the often cumbersome "on" language. It is also desirable tnat he dualize several of the proofs by writing out in order the duals of each proposition used in the proofs in question. EXERCISE Prove the theorem of duality for a space of four dimensions : Any propo- sition derivable from the assumptions of alignment and extension and closure for a space of four dimensions concerning points, lines, planes, and three- spaces remains valid when stated in the " on " terminology, if the words point and three-space and the words line and plane be interchanged. ' * 12. The theorems of alignment for a space of n dimensions. We have already called attention to the fact that Assumption E3',- whereby we limited ourselves to the consideration of a space of only * This section may be omitted on a first reading. 30 THEOEEMS OF ALIGNMENT AND DUALITY [CHAP. I three dimensions, is entirely arbitrary. This section is devoted to the discussion of the theorems of alignment, i.e. theorems derivable from Assumptions A and E 0, for a space of any number of dimensions. In this section, then, we make use of Assumptions A and E only. DEFINITION. If ^, P 19 -Z?, , P n are n + 1 points not on the same (n l)-space, and S n _ l is an ( l)-space on J%, P 2> - , P n) the class S n of all points such that every point of S n is on a line with J% and some point of S n _ 1 is called the n-space determined by J^ and S B-1 . As a three-space has already been defined, this definition clearly determines the meaning of "w-space" for every positive integral value of n. We shall use S n as a symbol for an %-space, calling a plane a 2-space, a line a 1 -space, and a point a 0-space, when this is convenient. S is then a symbol for a point. DEFINITION. An S r is on an S t and an S^ is on an S r (r<t) 9 pro- vided that every point of S r is a point of S t . DEFINITION. Jo points are said to be independent, if there is no S^g which contains them all. Corresponding to the theorems of J 6-9 we shall now establish the propositions contained in the following Theorems S n l, S n 2, S ra 3. As these propositions have all been proved for the case n = 3, it is sufficient to prove them on the hypothesis that they have already been proved for the cases n = 3, 4, - -, n 1 ; i.e. we assume that the propositions contained in Theorem S n _ 1 l, a, &, c, d, e, /have been proved, and derive Theorem S n l, #,,/ from them. By the prin- ciple of mathematical induction this establishes the theorem for any n. THEOREM S n l. Let the n-space S n "be defined ly the point R and the a. There is an n-space on any n + 1 independent points. 1). Any line on two points of S n has one point in common with R n _ a , and is on S n . c. Any S r (r < n) on r + 1 independent points of S n is on S n . d. Any S r (r < n) on r + 1 independent points of S n has an S rnl in common with R n _ lf provided all r + 1 points are not on R n _ r e. Any line I on two points of S n has one point in common with any S n _ I on S m . / -Sf .T and T n _ a (T not on T n _^ are any point and any (n l.)-space respectively of the n-space determined ly R and R w-1 , the latter n-space is the same as that determined ly T and T w _ r 12] SPACE OF JV DIMENSIONS 31 Proof, a. Let the n + 1 independent points be JJ, P v - - . , P n . Then the points JJ, P 2 , -, JJ are independent; for, otherwise, there would exist an S n _ 2 containing them all (definition), and this S n _ 2 with J? would determine an S a-1 containing all the points JJ, j, - , %, con- trary to the hypothesis that they are independent. Hence, by Theorem S n _ 1 l a, there is an S^ on the points JJ, JJ, , Jg; and this S H ^ with JJ determines an w-space which is on the points JJ, JJ, J?, ,.. &. If the line I is on R or R n _ 1? the proposition is evident from the definition of S n . If I is not on R or R n _ 1} let A and J5 be the given points of I which are on S n . The lines R ^ and R Q B then meet R^ in two points A! and B J respectively. The line I then meets the two lines J3'R , R ^'; and hence, by Assumption A3, it must meet the line A'B* in a point P which is on R n-1 by Theorem S K _ 1 15. To show that every point of I is on S n , consider the points A, A r , P. Any line joining an arbitrary point Q of I to R , meets the two lines PA and A A 1 , and hence, by Assumption A3, meets the third line A 1 P. But every point of AlP is on R n _ l (Theorem S n-1 l 6), and hence Q is, by definition, a point of S n . c. This may be proved by induction with respect to r. For r 1 it reduces to Theorem S B 1 J. If the proposition is true for r = k 1,' all the points of an S k on k + 1 independent points of S n are, by definition and Theorem S^l/, on lines joining one of these points to the points of the S k _ l determined by the remaining k points. But under the hypothesis of the induction this S k _ 1 is on S n , and hence, by Theorem S n l 6, all points of S k are on S n . d. Let r+1 independent" points of S tt be J, J?, - -, P r and let j?J be not on R n _ r Each of the lines JJ P k (k = 1, , r) has a point Q k in common with R n _ l (by S B 1 J). The points Q 19 Q z , - - -, Q ? . are inde- pendent; for if not, they would all be on the same S ? ._ 2 , which, together with J^, would determine an S,^ containing all the points PI (by S ri-1 l 6). Hence, by S r-1 l a, there is an S r _ : on j, Q*,- ,Q r which, by c, is on both S r and S n . e. We will suppose, first, that one of the given points is R . Let the other be A. By definition I then meets R u-1 in a point A', and, by 5,^15, in only one such point. If R is on S n _ 1} no proof is required for this case. Suppose, then, that R is not on S n _ 1 , and let C be any point of S n . r The line R (7 meets R^ in a point C r (by definition). By d, S n _ 1 has in common with R tt _! an (n 2)-space, S n _ 2 , and, by 32 THEOREMS OF ALIGNMENT AND DUALITY [CHAP. I Theorem S W _ 1 1^, this has in common with the line A f C f at least one point D f . All points of the line D f C are then on S a-1 , by S n-1 l b. Now the line I meets the two lines C'D f and CC r ; hence it meets the line CD ] (Assumption A3), and has at least one point on S n _ r We will now suppose, secondly, that both of the given points are distinct from R . Let them be denoted by A and J3, and suppose that R is not on S tt _ r By the case just considered, the lines R Q A and R Q B meet S n-1 in two points A! and B 1 respectively. The line Z, which meets R ^' and R Q B r must then meet A'B 1 in a point which, by Theorem S n-1 lJ, is on S B-1 . Suppose, finally, that R is on S tt-1 , still under the hypothesis that Z is not on R . By d, S n _ l meets R n-1 in an (n 2)-space Q rt _ 2 , and the plane R Z meets R B-1 in a line I 1 . By Theorem S n _ l ie 9 V and Q n _ 2 have in common at least one point P. Now the lines Z and R P are on the plane R Z, and hence have in common a point Q (by Theorem S a l e = Theorem 5). By S W-1 1 b the point Q is common to S n _ 1 and Z. /. Let the %-space determined by T and T n-1 be denoted by T w . Any point of T n is on a line joining T with some point of T n _ r Hence, by &, every point of T n is on S n . Let P be any point of S n distinct from T . The line T P meets T n-1 in a point, by e. Hence every point of S B is a point of T B . GOROLLAEY. On n + 1 independent points there is one and but one S n . This is a consequence of Theorem S n l a and S n l/. The formal proof is left as an exercise. THEOEEM S n 2. An S r and an S k having in common an S p , but not an S p+1 , are on a common S r+ ^._ p and are not both on the so/me s > if n<r + kp. Proof. If Jc~p, S k is on S r . If Jc >p, let JJ be a point on S k not on S 9 . Then 1J and S r determine an S r+1 , and P^ and S p an S |)+l> such that S p+l is contained in S r+1 and S t . If Is > p + 1, let P% be a point of S k not on S p+1 . Then P z and S r+1 determine an S r+2 , while J> and S p+1 determine an S^ +2 , which is on S r+2 and S r This process can be continued until there results an S^, containing all the points of S k . By Theorem S n l, Cor., we have i=sJcp. At this stage in the process we obtain an S r+ j.^ which contains both S r and S k . The argument just made shows that 1J, JJ, , P k _ p , together with any set Q v Q z> . . ., Q r+1 , of r + 1 independent points of S r , constitute 12] SPACE OF N DIMENSIONS S3 a set of r + k p + 1 independent points, each of which is either in S, or S A . If S r and S k were both on an S n , where n< r + & p, these could not be independent. THEOKEM S W 3. ^w S r and an S k contained in an S tt ar &0#i 00, the same S,. +jt _ w . Proof. If there were less than r + k n+1 independent points common to S, and S^ say r + kn points, they would, by Theorem S n 2, determine an S q , where q = r + k (r + k n 1) = n -f 1. Theorems S n 2 and S M 3 can be remembered and applied very easily by means of a diagram in which S M is represented by n + 1 points. Thus, if n = 3, we have a set of four points. That any two S 's have an S x in common corresponds to the fact that any two sets of three must have at least two points in common. In the general case a set of r + 1 points and a set of k + 1 selected from the same set of n + 1 have in common at least r + k n + 1 points, and this corresponds to the last theorem. This diagram is what our assumptions would describe directly, if Assumption E were replaced by the assumption: JSvery line contains two and only tivo points. If one wishes to confine one's attention to the geometry in a space of a given number of dimensions, Assumptions E 2, E 3, and E 3' may be replaced by the following : En. Not all points are on the same S XJ if k < n. En'. If S is an S n , all points are on S. Eor every S n there is a principle of duality analogous to that which, we have discussed for n = 3. In S n the duality is between S k and S n _ t _ l (counting a point as an.$ ), for all &'s from to n 1. If n is odd, there is a self-dual space in S n ; if n is even, S ff contains no self-dual space. EXERCISES 1. State and prove the theorems of duality in S 5 ; in S tt . 2. If m + 1 is the number of points on a line, how many S A /s are there in anS B ? * 3. State the assumptions of extension by which to replace Assumption En and En' for spaces of an infinite number of dimensions. Make use of the transfmite numbers. * Exercises marked * are of a more advanced or difficult character. CHAPTER II PROJECTION, SECTION, PERSPECTIVITY. ELEMENTARY CONFIGURATIONS 13. Projection, section, perspectivity. The point, line, and plane are the simple elements of space * ; we have seen in the preceding chapter that the relation expressed by the word on is a reciprocal relation that may exist between any two of these simple elements. In the sequel we shall have little occasion to return to the notion of a line as being a class of points, or to the definition of a plane ; but shall regard these elements simply as .entities for which the relation " on " has been denned. The theorems of the preceding chapter are to be regarded as expressing the fundamental properties of this relation.! We proceed now to the study of certain sets of these elements, and begin with a series of definitions. DEFINITION. A figure is any set of points, lines, and planes in space. A plane figure is any set of points and lines on the same plane. A point figure is any set of planes and lines on the same point. It should be observed that the notion of a point figure is the space dual of the notion of a plane figure. In the future we shall fre- quently place dual definitions and theorems side by side. By virtue of the principle of duality it will be necessary to give the proof of only one of two dual theorems. DEFINITION. Given a figure F DEFINITION. Given a figure F and a point P\ every point of F and a plane TT; every plane of F distinct from P determines with distinct from TT determines with P a line, and every line of F not TT a line, and every line of F not on P determines with P a plane; on TT determines with TT a point; the set of these lines and planes the set of these lines and points through P is called the projection on TT is called the section $ of F * The word space is used in place of the three-space in which are all the elements considered. t We shall not in future, however, confine ourselves to the " on " terminology, but shall also use the more common expressions. | A section by a plane is often called a plane section. 34 13] PEOJECTION, SECTION", PEESPECTIYITY 35 of F from P, The individual lines by TT. The individual lines and and planes of the projection are points of the section are also also called the projectors of the called the traces of the respective respective points and lines of F. planes and lines of F. If F is a plane figure and the point P is in the plane of the figure, the definition of the projection of F from P has the following plane dual : DEFINITION. Given a plane figure F and a line I in the plane of F; the set of points in which the lines of F distinct from I meet I is called the section of F by Z. The line I is called a transversal, and the points are called the traces of the respective lines of F. As examples of these definitions we mention the following: The projection of three mutually intersecting nonconcurrent lines from a point P not in the plane of the lines consists of three planes through P; the lines of intersection of these planes are part of the projection only if the points of intersection of the lines are thought of as part of the projected figure. The section of a set of planes all on the same line by a plane not on this line consists of a set of concurrent lines, the traces of the planes. The section of this set of concurrent lines in a plane by a line in the plane not on their common point consists of a set of points on the transversal, the points being the traces of the respective lines. DEFINITION. Two figures F 1? F 2 are said to be in (1, 1) correspond- ence or to correspond in a one-to-one reciprocal way, if every element of F.,^ corresponds (cf. footnote, p. 5) to a unique element of F 2 in such a way that every element of F 2 is the correspondent of a unique ele- ment of F r A figure is in (1, 1) correspondence with itself, if every element of the figure corresponds to a unique element of the same figure in such a way that every element of the figure is the corre- spondent of a unique element. Two elements that are associated in this way are said to be corresponding or homologous elements. A correspondence of fundamental importance is described in the following definitions: DEFINITION. If any two homol- DEFINITION. If any two homol- ogous elements of two corre- ogous elements of two corre- sponding figures have the same spending figures have the same projector from a fixed point 0, trace in a fixed plane o>, such such that all the projectors are that all the traces of either 36 PROJECTION", SECTION, PEESPECTIVITY [CHAP. II distinct, the figures are said to figure are distinct, the figures are be perspective from 0. The point said to be perspective from co, is called the center of perspec- The plane w is called the plane tivity. of perspectivity. DEFINITION. If any two homologous lines in two corresponding figures in the same plane have the same trace on a line I, such that all the traces of either figure are distinct, the figures are said to be perspective from 1. The line I is called the axis of perspectivity. Additional definitions of perspective 'figures will be given in the next chapter (p. 56). These are sufficient for our present purpose. DEFINITION. To project a figure in a plane a from a point onto a plane a r , distinct from a, is to form the section by cc r of the projection of the given figure from 0. To project a set of points of a line I from a point onto a line V 9 distinct from I but in the same plane with I and 0, is to form the section by l f of the projection of the set of points from 0. Clearly in either case the two figures are perspective from 0, pro- vided is not on either of the planes a, a 1 or the lines I, l r . EXERCISE What is the dual of the process described in the last definition ? The notions of projection and section and perspectivity are fun- damental in all that follows.* They will be made use of almost immediately in deriving one of the most important theorems of pro- jective geometry. We proceed first, however, to define an important class of figures. 14. The complete n-point, etc. DEFINITION. A complete n-point in space or a complete space n-point is the figure formed by n points, no four of which lie in the same plane, together with the n(n 1)/2 lines joining every pair of the points and the n(n l)(n 2)/6 planes joining every set of three of the points. The points, lines, and planes of this figure are called the vertices, edges, and faces respectively of the complete 7i-point * The use of these notions in deriving geometrical theorems goes back to early times. Thus, e.g., B. Pascal (1623-1662) made use of them in deriving the theorem on a hexagon inscribed in a conic which bears his name. The systematic treatment of these notions is due to Ponceletj cl his Traits des proprie'te's proiectives des figures, Paris, 1822. 14] JV-POINT, iV-PLANE, JV-LIXE 37 The simplest complete %-point in space is the complete space four-point. It consists of four vertices, six edges, and four faces, and is called a tetrahedron. It is a self-dual figure. EXERCISE Define the complete n-plane in space by dualizing the last definition. The planes, lines, and points of the complete n-plane are also called the faces, edges, and vertices of the n-plane. DEFINITION. A complete n-point in a plane or a complete plane n-point is the figure formed by n points of a plane, no three of which are collinear, together with the n(n 1)/2 lines joining every pair of the points. The points are called the vertices and the lines are called the sides of the n-point. The plane dual of a complete plane ^-point is called a complete plane n-line. It has n sides and n(n V)/2 vertices. The simplest complete plane n-point consists of three vertices and three sides and is called a triangle. DEFINITION. A simple space nyoint is a set of n points J^, J^, P v ,P n taken in a certain order, in which no four consecutive points are coplanar, together with the n lines J?J, JJJJ, , PJE( joining suc- cessive points and the n planes J%J%Z, -, J^^ determined by successive lines. The points, lines, and planes are called the vertices, edges, and faces respectively of the figure. The space dual of a simple space 7i-point is a simple space n-plane. DEFINITION. A simple plane n-point is a set of n points P^, P%, P%, - JJ of a plane taken in a certain order in which no three consecutive points are collinear, together with the n lines JJ.2J, P^, -, P n l^ joining suc- cessive points. The points and lines are called the vertices and sides respectively of the figure. The plane dual of a simple plane n-point is called a simple plane n-line. Evidently the simple space n-point and the simple space n-plane are identical figures, as likewise the simple plane n-point and the simple plane n-line. Two sides of a simple n-line which meet in one of its vertices are adjacent. Two vertices are adjacent if in the dual relation. Two vertices of a simple n-point J%P% - P n (n even) are opposite if, in the order J^ P n , as many vertices follow one and precede the other as precede the one and follow the other. If n is odd, a vertex and a side are opposite if, in the order .TJJfJ - P n , as many vertices follow the side and precede the vertex as follow the vertex and precede the side. 38 PROJECTION, SECTION, PEESPECTIVITY [CHAP.II The space duals of the complete plane %-point and the complete plane %-line are the complete n-plane on a point and the complete n-line on a point respectively. They are the projections from a point, of the plane ^-line and the plane ^-point respectively. 15. Configurations. The figures defined in the preceding section are examples of a more general class of figures of which we will now give a general definition. DEFINITION. A figure is called a configuration, if it consists of a finite number of points, lines, and planes, with the property that each point is on the same number & 12 of lines and also on the same num- ber a ia of planes ; each line is on the same number & 21 of points and the same number # 23 of planes ; and each plane is on the same number a 81 of points and the same number a 32 of lines. A configuration may conveniently be described by a square matrix : 1 point 2 line 3 plane 1 point 2 line 3 plane 31 32 33 In this notation, if we call a point an element of the first kind, a line an element of the second kind, and a plane one of the third kind, the number a (i = j) gives the number of elements of the /th kind on every element of the ith kind. The numbers a iv a 22t a ss give the total number of points, lines, and planes respectively. Such a square matrix is called the symbol of the configuration. A tetrahedron, for example, is a figure consisting of four points, six lines, and four planes ; on every line of the figure are two points of the figure, on every plane are three points, through every point pass three lines and also three planes, every plane contains three lines, and through every line pass two planes. A tetrahedron is therefore a configuration of the symbol COlSTFIG-UBATIOlSrS 39 The symmetry shown in this symbol is due to the fact that the figure in question is self-dual. A triangle evidently has the symbol Since all the numbers referring to planes are of no importance in case of a plane figure, they are omitted from the symbol for a plane configuration. In general, a complete plane ?i-point is of the symbol n 2 w-1 and a complete space w-point of the symbol 2 Jn(n-l) w-2 3 3 t n (n-l)(a-2) Further examples of configurations are figs. 14 and 15, regarded as plane figures. EXERCISE Prove that the numbers in a configuration symbol must satisfy the condition (* / = 1, 2, 8) 16. The Desargues configuration. A very important configuration is obtained by taking the plane section of a complete space five-point. The five-point is clearly a configuration with the symbol and it is clear that the section by a plane not on any of the vertices is a configuration whose symbol may be obtained from the one just given by removing the first column and the first row. This is due to the fact that every line of the space figure gives rise to a point in 40 PBO JECTION, SECTION, PEESPECTIYITY [CHAP, n the plane, and every plane gives rise to a line. The configuration in the plane has then the symbol We proceed to study in detail the properties of the configuration just obtained. It is known as the configuration of Desargues. We may consider the vertices of the complete space five-point as con- sisting of the vertices of a triangle A, B 9 C and of two points O l} 2 Fio. 14 not coplanar with any two vertices of the triangle (fig. 14). The sec- tion by a plane a not passing through any of the vertices will then consist of the following : A triangle A^B^C^ the projection of the triangle ABC from : on a. A triangle J 2 2? 2 <7 2 , the projection of the triangle ABC from 2 on a. The trace of the line 0^0 y The traces A zi B^ <7 3 of the lines BC, CA, AB respectively. The trace of the plane ABC, which contains the points A z , B^ C y The traces of the three planes AO^O^ BO^O^ COft^, which contain respectively the triples of points OA^A^ OB^B^ OC^C y The configuration may then be considered (in ten ways) as consist- ing of two triangles A^B^ and A^B^C V perspective from a point and THEOREM OF DESAEGUES 41 having homologous sides meeting in three collinear points J 3 , B B , <7 8 . These considerations lead to the following fundamental theorem: . THEOREM 1. THE THEOREM OF DESARGUES.* If two triangles in the same plane are 'perspective "from a point, the three pairs of homologous sides meet in collinear points; i.e. the triangles are perspective from a line. '(A, E) Proof. Let the two triangles be A^B^ and A 2 J3 Z C 2 (fig. 14), the lines A^A^ B^^ ^aA meeting in the point 0. Let B^A V B^A 2 inter- sect in the point <? 3 ; A^C V A 2 C Q in B^\ B^C V B Z C Z in A z . It is required to prove that A z , B B , <? s are coUinear, Consider any line through which is not in the plane of the triangles, and denote by O v 3 any two distinct points on this line other than 0. Since the lines A 2 Z and A^O^ lie in the plane (A^A^ 1 2 ), they intersect in a point A. Similarly, B^O^ and J? 2 2 intersect in a point B, and likewise C^ and C 2 2 in a point C. Thus ABCO^O^ together with the lines and planes determined by them, form a complete five-point in space of which the perspective triangles form a part of a plane section. The theorem is proved by completing the plane section. Since AB lies in a plane with AJ& V and also in a plane with A Z B 2 , the lines AJ& V A Z B 2) and AB meet in 9 . So also A 1 C V A Z C Z) and AC meet in B B ; and B t C v B Z C 2 , and BC meet in A y Since A z) B z , C s lie in the plane ABC and also in the plane of the triangles A^B^C^ and A^B^C^ they are collinear. THEOREM l f . If two triangles in the same plane are perspective from a line, the lines joining pairs of homologous vertices are, con- current; i.e. the triangles are perspective from a point. (A, E) This, the converse of Theorem 1, is also its plane dual, and hence requires no further proof. COROLLARY. If two triangles not in the same plane are perspective from a point, the pairs of homologous sides intersect in collinear points; and conversely. (A, E) A more symmetrical and for many purposes more convenient nota- tion for the Desargues configuration may be obtained as follows: Let the vertices of the space five-point be denoted by ^, J^, P t) P^ J (fig. 15). The trace of the line P^ in the plane section is then ' naturally denoted by ^ in general, the trace of the line JJJ^ by 7?. (i, j = 1, 2, 3, 4, 5, i = /). Likewise the trace of the plane j?J^5 ma 7 * G-irard Desargues, 1593-1662. 42 PKQJECTICH8", SECTION, PER8PECTIVITY [CHAP, n be denoted by l vt (i, j, k = 1, 2, 3, 4, 5). This notation makes it pos- sible to te]l at a glance which lines and points are united. Clearly a point is on a line of the configuration if and only if the suffixes of the point are both among the suffixes of the line. Also the third point on the line joining JJ and J^ is the point JJ, ; two points are on the same line if and only if they have a suffix in common, etc. EXERCISES 1. Prove Theorem V without making use of the principle of duality. 2. If two complete w-points in different planes are perspective from a point, the pairs of homologous sides intersect in collinear points. "What is the dual theorem ? What is the corresponding theorem concerning any two plane figures in. different planes ? 3. State and prove the converse of the theorems in Ex. 2. 4. If two complete ra-points in the same plane correspond in such a way that homologous sides intersect in points of a straight line, the lines joining homologous vertices are concurrent ; i.e. the two w-points are perspective from a point. Dualize. 5. What is the figure formed by two complete n-points in the same plane when they are perspective from a point? Consider particularly the cases n = 4 and n = 5. Show that the figure corresponding to the general case is a plane section of a complete space (n + 2)-point. Give the configuration symbol and dualize. 6. If three triangles are perspective from the same point, the three axes of perspectivity of the three pairs of triangles are concurrent ; and conversely. Dualize, and compare the configuration of the dual theorem with the case n = 4 of Ex. 5 (cf. fig. 15, regarded as a plane figure). 17] PEESPECTIVE TETEAHEDEA 43 17. Perspective tetrahedra. As an application of tlie corollary of the last theorem we may now derive a theorem in space analogous to the theorem of Desargues in the plane. THEOREM 2. If two tetrahedra are perspective from a point, tlie six p airs of homologous edges intersect in coplanar points, and the four pairs of homologous faces intersect in coplanar lines ; i.e. the tetra- are perspective from a plane. (A, E) :*' FIG. 16 Proof. Let the two tetrahedra be -Zy^J-ZJ and JJ'JJ'JJ'JEJ', and let the lines JJJJ', J^', jZ^', JJ-Z^ meet in the center of perspectivity 0. Two homologous edges P % P 3 and J?'J' then clearly intersect ; call the point of intersection P v . The points ^ 2 , J? s , JJ S lie on the same line, since the triangles JJJSJJ and -ZJ'Ia'JJ' are perspective from (The- orem 1, Cor.). By similar reasoning applied to the other pairs of perspective triangles we find that the following triples of points are collinear: The first two triples have the point P^ in common, and hence determine a plane;, each of the other two triples has a point in 44 PKOJECTION, SECTION, PEESPECTIYITY [CHAP.II common with each, of the first two. Hence all the points P %1 lie in the same plane. The lines of the four triples just given are the lines of intersection of the pairs of homologous faces of the tetrahedra. The theorem is therefore proved. THEOREM 2'. If two tetrahedra are perspective from a plane, the lines joining pairs of homologous vertices are concurrent, as likewise the planes determined ly pairs of homologous edges ; i.e. the tetrahedra are perspective from a point (A, E) This is the space dual and the converse of Theorem 2. EXERCISE "Write the symbols for the configurations of the last two theorems. 18. The quadrangle-quadrilateral configuration. DEFINITION. A complete plane DEFINITION. A complete plane four-point is called a complete four-line is called a complete quadrangle. It consists of four quadrilateral It consists of four vertices and six sides. Two sides sides and six vertices. Two ver- not on the same vertex are called tices not on the same side are opposite. The intersection of two called opposite. The line joining opposite sides is called a diag- two opposite vertices is called a onal point. If the three diagonal diagonal line. If the three diag- points are not collinear, the tri- onal lines are not concurrent, the angle formed by them is called triangle formed by them is called the diagonal triangle of the the diagonal triangle of the quadrangle.* quadrilateral* The assumptions A and E on which all our reasoning is based do not suffice to prove that there are more than three points on any line. In fact, they are all satisfied by the triple system (1), p. 3 (cf. fig. 17). In a case like this the diagonal points of a complete quadrangle, ,ajce ^collinear.and.the di^onSTlmes of sT^qm^ete quadrilateral concur- rent^ as may readily be verified. Two perspective triangles cannot exist in such a plane, and hence the Desargues theorem becomes * In general, the intersection of two sides of a complete plane w-point which do not have a vertex in common is called a diagonal poM of the ra-point, and the line joining two vertices of a complete plane n-line which do not lie on the same side is called a diagonal line of the 7i-line. A complete plane n~point (n-line) then has n(n - 1) (n - 2) (n -3)/8 diagonal points (lines). Diagonal points and lines are sometimes called false vertices and false sides respectively. ifl ASSUMPTION H trivial. Later on we shall add an assumption* which excludes all such cases as this, and, in fact, provides for the existence of an in- finite number of points on a line. A part of what is contained in this assumption is the following: ASSUMPTION H . Tlie diagonal points of a complete quadrangle are noncollinear. Many of the important theorems of geometry, however, require the existence of no more than a finite number of points. "We shall there- fore proceed without the use of w P IG 17 further assumptions than A and E, understanding that in order to give our theorems meaning there must ~be postulated, the existence of the points specified in their hypotheses. In most cases the existence of a sufficient number of points is insured by Assumption H , and the reader who is taking up the subject for the first time may well take it as having been added to A and E. It is to be used in the solution of problems. We return now to a further study of the Desargues configuration. A complete space five-point may evidently be regarded (in five ways) as a tetrahedron and a complete four-line at a point A plane section of a four-line is a quadrangle and the plane section of a tetrahedron is a quadrilateral. It follows that (in five ways) the Desargues con- figuration may be regarded as a quadrangle and a quadrilateral. Moreover, it is clear that the six sides of the quadrangle pass through the six vertices of the quadrilateral. In the notation described on page 41 one such quadrangle is JJ 2 , ^ 3 , JJ 4 , JJ 5 and the corresponding quadrilateral is Z 234 , Z 235 , ? 2 45> Z 845 . The question now naturally arises as to placing the figures thus ob- tained in special relations. As an application of the theorem of De- sargues we will show how to construct f a quadrilateral which has the same diagonal triangle as a given quadrangle. "We will assume in our discussion that the diagonal points of any quadrangle form a triangle. * Merely saying that there are more than three points on a line does not insure that the diagonal points of a quadrangle are noncollinear. Cases where the diagonal points are collinear occur whenever the number of points on a line is 2 + ! t To construct a figure is to determine its elements in terms of certain given elements. 46 PROJECTION, SECTION", PEKSPECTIVITY [CHAP, n Let PV P^ p *> % be the vertices of the given complete quadrangle, and let D 12 , Z> 13 , D u be the vertices of the diagonal triangle, D 12 being on the side Pf^ > 1B on the side '%P B , and D 14 on the side JJJJ (fig. 18). We observe first that the diagonal triangle is perspective with each of the four triangles formed ly a set of three of the vertices of the quadrangle, the center of perspectivity being in each case the fourth vertex. This gives rise to four axes of perspectivity (Theorem 1), one corresponding to each vertex of the quadrangle.* These four lines clearly form the sides of a complete quadrilateral whose diagonal triangle is Z> 12 , D 13 , Z> 14 . It may readily be verified, by selecting two perspective triangles, that the figure just formed is, indeed, a Deja^^ This special case of the Desargues configuration is called the qruadrangle- qiiadrilateral configuration.^ EXERCISES 1. If p is the polar of P with regard to the triangle ABC, then P is the pole of p with regard to the same triangle ; that is, P is obtained from p by a construction dual to that used in deriving p from P. From this theorem it follows that the relation between the quadrangle and quadrilateral in this * The line thus uniquely associated with a vertex is called the polar of the point with respect to the triangle formed by the remaiDing three vertices. The plane dual process leads to a point associated with any line. This point is called the pole of the line with respect to the triangle. t A further discussion of this configuration and its generalizations will be found in the thesis of H. F. McNeish. Some of the results in this paper are indicated in the exercises. 18,19] QUADBAjSTGULAK SETS 47 configuration is mutual ; that is, if either is given, the other is determined. For a reason ^rhich will be evident later, either is called a covariant of the other. 2. Show that the configuration consisting of two perspective tetrahedra, their center and plane of perspectivity, and the projectors and traces may be regarded in six ways as consisting of a complete 5-point P 12 , P 13 , P u , P 15 , P 16 and a complete 5-plane ir S456 , *- 3456 , 7r 233G , 7r 2S46 , 7r 234 -, the notation being analogous to that used on page 41 for the Desargues configuration. Show- that the vertices of the 5-plane are on the faces of the 5-point. 3. If P 15 P 2 , P g , P 4 , P 5 , are vertices of a complete space 5-point, the ten points Dy, in which an edge p {j meets a face P l P l P m (i,/, fc, I, m all distinct), are called diagonal points. The tetrahedra P 2 P 8 P 4 P 5 and -Z^-^is^w^iB are per- spective with P x as center. Their plane of perspectivity, v 19 is called the polar of P x with regard to the four vertices. In like manner, the points P 2 , P 3 , P 4 , P 5 determine their polar planes 7r 2 , 7r 3 , 7r 4 , ir e . Prove that the 5-point and the polar 5-plane form the configuration of two perspective tetrahedra ; that the plane section of the 5-point by any of the five planes is a quadrangle-quadrilateral configuration ; and that the dual of the above construction applied to the 5-plane determines the original 5-point. 4. If P is the pole of IT with regard to the tetrahedron A^A 2 A Z A^ then is v the polar of P with regard to the same tetrahedron ? 19. The fundamental theorem on quadrangular sets. THEOREM 3. If two complete quadrangles %%J%% and P^PjPjPl correspond P^ to P^, 1% to P^ etc. in suc7i a way that five of the pairs of homologous sides intersect in points of a line I, then the sixth pair of homologous sides will intersect in a point of I. (A, E) This theorem holds whether the quadrangles are in the same or in different planes. Proof. Suppose, first, that none of the vertices or sides of one of the quadrangles coincide with any vertex or side of the other. Let %Pv %J%, -ZJ-Z2, JJJJ, P^PI be the five sides which, by hypothesis, meet their homologous sides P^Pj, P^Pj, PJPJ, PJPJ, P^Pj in points of I (fig. 19). We must show that P Z P^ and IgP meet in a point of I. The triangles ^P^ and P^P* are, by hypothesis, perspec- tive from l\ as also the triangles P^P and P^P^P^. Each pair is therefore (Theorem I/) jpersgective from appoint, and this point is in each case the intersection^^ of the lines JJZJ 7 and P^Ef. Hence the triangles P^P^ and ^^^ are perspective from and their pairs of homologous sides intersect in the points of a line, which is evi- dently I, since it contains two points of L But P^ and $P are 48 PBOJECTIOST, SECTION, PEESPECTIYITY [CHAP.H two homologous sides of these last two triangles. Hence they inter- sect in a point of the line I. If a vertex or side of one quadrangle coincides with a vertex or side of the other, the proof is made by considering a third quadrangle* whose vertices and sides are distinct from those of both of the others, and which has five of its sides passing through the five given points of intersection of homologous sides of the two given quadrangles. By the argument above, its sixth side will meet the sixth side respectively of each of the two given quadrangles in the same point of I. This completes the proof of the theorem. 1. It should be noted that the theorem is still valid if the line I con- tains one or more of the diagonal points of the quadrangles. The case in which I contains two diagonal points is of particular importance and will be discussed in Chap. IV, 31. NOTE 2. It is of importance to note in how far the quadrangle P^P^P^P^ is determined when the quadrangle P i P 2 P 8 P 4 and the line I are given. It may be readily verified that in such a case it is possible to choose any point P{ to correspond to any one of the vertices P t , P 2 , P 8 , P 4 , say P x ; and that if m is any line of the plane IP{ (not passing through Pf ) which meets one of the sides, say a, of P^PJP^P^ (not passing through P x ) in a point of I, then m may be chosen as the side homologous to a. But then the remainder of the figure is uniquely determined. * This evidently exists whenever the theorem is not trivially obvious. 19 ] QUADBAKG-ULAK SETS 49 THEOREM 3'. If two complete quadrilaterals a^a^ and a(a^a(a[ correspond a l to a[, 3 to a! 2) etc. in such a way that five of the lines joining homologous vertices pass through a point P, the line joining the sixth pair of homologous vertices will also pass through P. (A, E) This is the plane dual of Theorem 3 regarded as a plane theorem. DEFINITION. A set of points in which the sides of a complete quad- rangle meet a line I is called a quadrangular set of points. Any three sides of a quadrangle either, form a triangle or meet in a vertex ; in the former case they are saifl. to form a triangle triple, in the latter > point triple of lines. In a quadrangtilar set of points on a line I any three points in which the lines of a triangle triple meet I is called a triangle triple of points in the set ; three points in which the lines of a point triple meet I are called a point triple of points. A quadrangular set of points will be denoted by Q(ABC, DBF), where ABC is a point triple and DEF is a triangle triple, and where A and D, B and E, and and F are respectively the inter- sections with the line of the set of the pairs of opposite sides of the quadrangle. The notion of a quadrangular set is of great importance in much that follows. It should be noted again in this connection that one or two * of the pairs A, D or B, U or (7, F may consist of coincident points ; this occurs when the line of the set passes through one or two of the diagonal points.f We have just seen (Theorem 3) that if we have a quadrangular set of points obtained from a given quadrangle, there exist other quadrangles that give rise to the same quadrangular set In the quadrangles mentioned in Theorem 3 there corresponded to every triangle triple of one a triangle triple of the other. DEFINITION. When two quadrangles giving rise to the same quadrangular set are so related with reference to the set that to a triangle triple of one corresponds a triangle triple of the other, the * All three may consist of coincident points in a space in which the diagonal points of a complete quadrangle are collinear. t It should he kept in mind that similar remarks and a similar definition may he made to the effect that the lines joining the vertices of a quadrilateral to a point P form a quadrangular set of lines, etc. (cf. 30, Chap. IV). 50 PKOJECTION, SECTION, PEBSPECTIVITY [CHAP, n quadrangles are said to be similarly placed (fig. 20); if a point triple of one corresponds to a triangle triple of the other, they are said to be oppositely placed (fig. 21). It will be shown later (Chap. IV) that quadrangles oppositely placed with respect to a quadrangular set are indeed possible. FIG. 20 FIG. 21 With the notation for quadrangular sets defined above, the last theorem leads to the following COROLLARY. If all "but one of the points of a quadrangular set Q (ABC, DBF) are given, the remaining one is uniquely determined. (A, E) For two quadrangles giving rise to the same quadrangular set with the same notation must be similarly placed, and must hence be in correspondence as described in -the theorem. 19,20] DESAEGUES CONFIGURATION 51 The quadrangular set which is the section by a 1-space of a complete 4-point in a 2-space, the Desargues configuration which is the section by a 2-space of a complete 5-point in a 3-space, the configuration of two perspective tetra- hedra which maybe considered as the section by a 3-space of a complete 6-point in a 4-space are all special cases of the section by an n-space of a complete (n + 3)-point in an (n + l)-space. The theorems which we have developed for the three cases here considered are not wholly parallel. The reader will find it an entertaining and far from trivial exercise to develop the analogy in full. EXERCISES 1. A necessary and sufficient condition that three lines containing the ver- tices of a triangle shall be concurrent is that their intersections P, Q, It with a line I form, with intersections E, F, G of corresponding sides of the triangle with I, a quadrangular set Q(PQR, EFG).' 2. If on a given transversal line two quadrangles determine the same quad- rangular set and are similarly placed, their diagonal triangles are perspective from the center of perspectivity of the two quadrangles. 3. The polars of a point P on a line I with regard to all triangles which meet I in three fixed points pass through a common point P' on L 4. In a plane TT let there be given a quadrilateral a 19 2 , a s , a 4 and a point not on any of these lines. Let A 19 A 2 , A%, A 4 be any tetrahedron whose four faces pass through the lines a v 2 , a 3 , a respectively. The polar planes of with respect to all such tetrahedra pass through the same line of TT. 20. Additional remarks concerning the Desargues configuration. The ten edges of a complete space five-point may be regarded (in six ways) as the edges of two simple space five-points. Two such five-points are, for example, JJjZyjJJJj and P^P^P^. Corresponding thereto, the Desargues configuration may be regarded in six ways as a pair of simple plane pentagons (five-points). In our previous notation the two corresponding to the two simple space five-points just given are ^J^^^^i an< i -^is-^B^a^L^Jr Every vertex* of each of these pentagons is on a side of the other. Every point, P^ for instance, has associated with it a unique line of the configuration, viz. 1 M5 in the example given, whose notation does not contain the suffixes occurring in the notation of the point. The line may be called the polar of the point in the configuration, and the point the pole of the line. It is then readily seen that the polar of any point is the axis of perspectivity of two triangles whose center of perspectivity is the point. In case we regard the configuration as consisting of a complete quadrangle and complete 52 PEOJBCTIOK, SECTION, PERSPECTIVITY [CHAP, n quadrilateral, it is found that a pole and polar are homologous vertex and side of the quadrilateral and quadrangle. If we consider the configuration as consisting of two simple pentagons, a pole and polar are a vertex and its opposite side, e.g. ^ and -? 4 ^ 5 . The Desargues configuration is one of a class of configurations having similar properties. These configurations have been studied by a number of writers.* Some of the theorems contained in these memoirs appear in the exercises below. EXERCISES In discussing these exercises the existence should be assumed of a sufficient number of points on each line so that the figures in question do not degenerate. In some cases it may also be assumed that the diagonal points of a complete quadrangle are not cottinear. Without these assumptions our theorems are true, indeed, but trivial. 1. What is the peculiarity of the Desargues configuration obtained as the section of a complete space five-point by a plane which contains the point of intersection of an edge of the five-point with the face not containing this edge ? also by a plane containing two or three such points ? 2. Given a simple pentagon in a plane, construct another pentagon in the same plane, whose vertices lie on the sides of the first and whose sides con- tain the vertices of the first (cf. p. 51). Is the second uniquely determined when the first and one side of the second are given? S, 3. If two sets of three points A, B, C and A', B' } C' on two coplanar lines Z and V respectively are so related that the lines A A', BB', CC f are concurrent, then the points of intersection of the pairs of lines AB f and BA', BC' and CB', CA' and A C' are collinear with the point ZZ'. The line thus determined is called the polar of the point (A A', BB') with respect to I and Z'. Dualize. 4. Using the theorem of Ex. 3, give a construction for a line joining any given point in the plane of two lines Z, V to the point of intersection of Z, V without making use of the latter point. 5. Using the definition in Ex. 3, show that if the point P' is on the polar p of a point P with respect to two lines Z, Z', then the point P is on the polar p' of P' with respect to Z, V. 6. If the vertices A 19 A^ -4 3 , A of a simple plane quadrangle are respec- tively on the sides a z , a 3 , a s , a 4 of a simple plane quadrilateral, and if the inter- section of the pair of opposite sides A^A^ A%A is on the line joining the pair of opposite points a^, <z 2 a 8 , the remaining pair of opposite sides of the quad- rangle will meet on the line joining the remaining pair of opposite vertices of the quadrilateral. Dualize. * A. Cayley, Collected Works, Vol. I (1846), p. 317. G. Veronese, Mathema- tische Annalen, Vol. XIX (1882). Further references will be found in a paper by W. B. Carver, Transactions of the American Mathematical Society, Vol. VI (1905), p. 534. 20] EXEECISES 53 7. If two complete plane n-points A^ A 2 , - , A n and A{, A^ , A' n are so related that the side A^A^ and the remaining 2 (n 2) sides passing through ^ and ^L 2 meet the corresponding sides of the other rc-point in points of a line Z, the remaining pairs of homologous sides of the two n -points meet on I and the two n-points are perspective from a point. Dualize. 8. If five sides of a complete quadrangle A^A^A^A^ pass through five vertices of a complete quadrilateral a 1 a a a 8 a 4 in such a way that A^A^ is on <2 3 a 4 , A 2 A S on a 4 a 1? etc., then the sixth side of the quadrangle passes through the sixth vertex of the quadrilateral. Dualize. 9. If on each of three concurrent lines a, &, c two points are given, A lt A 3 on a; jB 1? B 2 on &; C v C 2 on c, there can be formed four pairs of triangles AfBjC L (i,j, k = 1, 2) and the pairs of corresponding sides meet in six points which are the vertices of a complete quadrilateral (Veronese, Atti dei Lincei, 1876-1877, p. 649). 10. With nine points situated in sets of three on three concurrent lines are formed 36 sets of three perspective triangles. For each set of three dis- tinct triangles the axes of perspectivity meet in a point; and the 36 points thus obtained from the 36 sets of triangles lie in sets of four on 27 lines, o/> q giving a configuration (Veronese, loc. cit.). 11. A plane section of a 6-point in space can be considered as 3 triangles perspective in pairs from 3 collinear points with corresponding sides meeting in 3 collinear points. 12. A plane section of a 6-point in space can be considered as 2 perspective complete quadrangles with corresponding sides meeting in the vertices of a complete quadrilateral. 13. A plane section of an n-point in space gives the configuration * which may be considered (in n C n _ k ways) as a set of (n ) fc-points perspective in pairs from n _x,C a points, which form a configuration the points of intersection of corresponding sides form a coi _*A n fc 2 & " n-*^ an< ifiguration /A fc-2 3 k C s 14. A plane section of a 7-point in space can be considered (in 120 ways) as composed of three simple heptagons (7-points) cyclically circumscribing each other. 15. A plane section of an 11-point in space can be considered (in 19 ways) as composed of four 11-points cyclically circumscribing each other. 16. A plane section of an n-point in space for n prime can be considered (in [n 2 ways) as n ~ simple n-points cyclically circumscribing each other. 2 * The symbol n C r is used to denote the number of combinations of n things taken r at a time. 54 PROJECTION, SECTION, PERSPECTIVES [CHAP, n 17. A plane section of a 6-point in space gives (in six ways) a 5-point whose sides pass through the points of a configuration 18. A plane section of an n -point in space gives a complete (n 1) -point whose sides pass through the points of a configuration * 19. The n-space section of an ??2-point (m z. n + 2) in an (n + l)-space can be considered in the w-space as (m &) ^-points (in m (? m _ x ways) perspective in pairs from the vertices of the ?i-space section of one (m I')-point ; the r-spaces of the /j-point figures meet in (r 1) -spaces (r = l,2, ,w 1) which form the rc-space section of a -point. * 20. The figure of two perspective (n + l)-points in an w-space separates (in n + 3 ways) into two dual figures, respectively an (n + 2)-point circum- scribing the figure of (n + 2) (n l)-spaces. * 21. The section by a 3-space of an n-point in 4-space is a configuration M c, 71-2 -2<?2 3 M C 3 n-3 6 4 ,A The plane section of this configuration is 22. Let there be three points on each of two concurrent lines l v / 2 . The nine lines joining points of one set of three to points of the other determine six triangles whose vertices are not on / t or l z . The point of intersection of ^ and Z 2 has the same polar with regard to all six of these triangles. 23. If two triangles are perspective, then are perspective also the two triangles whose vertices are points of intersection of each side of the given triangles with a line joining a fixed point of the axis of perspectivity to the opposite vertex. *24. Show that the configuration of the two perspective tetrahedra of Theorem 2 can be obtained as the section by a 3-space of a complete 6-point in a 4-space. *25. If two 5-points in a 4-space are perspective from a point, the corre- sponding edges meet in the vertices, the corresponding plane faces meet in the lines, and the corresponding 3-space faces in the planes of a complete 5-plane in a 3-space. * 26. If two (n -f l)-points in an ra-space are perspective from a point, their corresponding r-spaces meet in (r 1) -spaces which lie in the same (n l)-space (r=rl, 2 , 1) and form a complete configuration of (n + 1) (n 2) -spaces in (n 1) -space. CHAPTER III PROJECTIVITIES OF THE PRIMITIVE GEOMETRIC FORMS OF ONE, TWO, AND THREE DIMENSIONS 21. The nine primitive geometric forms. DEFINITION. A. pencil of points DEFINITION. A. pencil of planes or a range is the figure formed by or an axial pencil * is the figure the set of all points on the same formed by the set of all planes on line. The line is called the axis the same line. The line is called of the pencil. the axis of the pencil. As indicated, the pencil of points is the space dual of the pencil of planes. ^/ DEFINITION. A pencil of lines or a flat pencil is the figure formed by the set of all lines which are at once on the same point and the same plane ; the point is called the vertex or center of the pencil The pencil of lines is clearly self-dual in space, while it is the plane dual of the pencil of points. The pencil of points, the pencil of lines, and the pencil of planes are called the primitive geometric forms of the first grade or of one dimension. DEFINITION. The following are known as the primitive geometric forms of the second grade or of two dimensions : The set of all points on a plane The set of all planes on a point is called a plane of points. The is called a bundle of planes. The set of all lines on a plane is called set of all lines on a point is called a plane of lines: The plane is a bundle of lines . The point is called the "base of the two forms, called the center of the bundles. The figure composed of a plane The figure composed of a bundle of points and a plane of lines of lines and a bundle of planes with the same base is called a with the same center is called planar field. simply a "bundle. DEFINITION. The set of all planes in space and the set of all points' in space are called the primitive geometric forms of the third grade or of three dimensions. * The pencil of planes is also called "by some writers a sheaf. 55 56 PRIMITIVE GEOMETRIC FORMS [CHAP, in There are then, all told, nine primitive geometric forms in a space of three dimensions.* 22. Perspectivity and projectivity. In Chap. II, 13, we gave a definition of perspectivity. This definition we will now apply to the case of two primitive forms and will complete it where needed. We note first that, according to the definition referred to, two pencils of points in the same plane are perspective provided every two homol- ogous points of the pencils are on a line of a flat pencil, for they then have the same projection from a point. Two ^planes of points (lines) are perspective, if every two homologous elements are on a line (plane) of a bundle of lines (planes). Two pencils of lines in the same plane are perspective, if every two homologous lines intersect in a point of the same pencil of points. Two pencils of planes are perspective, if every two homologous planes are on a point of a pencil of points (they then have the same section by a line). Two bundles of lines (planes) are perspective, if every two homologous lines (planes) are on a point (line) of a plane of points (lines) (they then have the same section by a plane), etc. Our previous definition does not, how- ever, cover all possible eases. In the first place, it does not allow for the possibility of two forms of different kinds being perspective, such as a pencil of points and a pencil of lines, a plane of points and a bundle of lines, etc. This lack of completeness is removed for the case of one-dimensional forms by the following definition. It should be clearly noted that it is in complete agreement with the previous definition of perspectivity ; as far as one-dimensional forms are con- cerned it is wider in its application. DEFINITION. Two one-dimensional primitive forms of different kinds, not having a common axis, are perspective, if and only if they corre- spond in such a (1, 1) way that each element of one is on its homol- ogous element in the other ; two one-dimensional primitive forms of the same kind are perspective, if and only if every two homologous elements are on an element of a third one-dimensional form not having an axis in common with one of the given forms. If the third form is a pencil of lines with vertex P, the perspectivity is said to be * Some writers enumerate only six, by defining the set of all points and lines on a plane as a single form, and by regarding the set of all planes and lines at a point and the set of all points and planes in space each as a single form. We have fol- lowed the usage of Enriques, Vorlesungen iiber Projektive Geometric. 22] PEESPECTIYITY 57 central with center P; if the third form is a pencil of points or a pencil of planes with axis Z, the perspectivity is said to be axial with axis Z. As examples of this definition we mention the following: Two pencils of points on skew lines are perspective, if every two homol- ogous elements are on a plane of a pencil of planes ; two pencils of lines in different planes are perspective, if every two homologous lines are on a point of a pencil of points or a plane of a pencil of planes (either of the latter conditions is a consequence of the other) ; two pencils of planes are perspective, if every two homologous planes are on a point of a pencil of points or a line of a pencil of lines (in the latter case the axes of the pencils of planes are coplanar). A pen- cil of points and a pencil of lines are perspective, if every point is on its homologous line, etc. It is of great importance to note that our definitions of perspective primitive forms are dual throughout; ie. that if two forms are per- spective, the dual figure will consist of perspective forms. Hence any theorem proved concerning perspectivities can at once be dualized ; in particular, any theorem concerning the perspectivity of two forms of the same kind is true of any other two forms of the same kind. We use the notation [P] to denote a class of elements of any kind and denote individuals of the class by P alone or with an index or subscript. Thus two ranges of points may be denoted by [P] and [Q]. To indicate a perspective correspondence between them we write The same symbol, ^, is also used to indicate a perspectivity between any two one-dimensional forms. If the two forms' are of the same kind, it implies that there exists a third form such that every pair of homologous elements of the first two forms is on an element of the third form. The third form may also be exhibited in the notation by placing a symbol representing the third form immediately over the sign of perspectivity, ^. Thus the symbols denote that the range [P] is perspective by means of the center A with the range [Q], that each Q is on a line r of the flat pencil [>], and that the pencil [r] is perspective by; the axis a with the flat pencil [*]. 58 PEIMITIVE GEOMETRIC FOEMS [CHAP, in A class of elements containing a finite number of elements can be indicated by the symbols for the several elements. When this notation is used, the symbol of perspectivity indicates that elements appearing in corresponding places in the two sequences of symbols are homologous. Thus 123 4: = AJ3C D A implies that 1 and A, 2 and B, 3 and <7, 4 and D are homologous. DEFINITION.* Two one-dimensional primitive forms [cr] and [V] (of the same or different kinds) are said to be projective, provided there exists a sequence of forms [r], |V] ? , [r (n) ] such that [<r] == [r] = [r'] = -== [T ( >] = [>']. . LJ A LJ A L J A A L J A L J The correspondence thus established between [cr] and [V] is called a protective correspondence or projectivity , or also a protective trans- formation. Any element cr is said to be projected into its homologous element cr' by the sequence of perspectivities. Thus a projectivity is the resultant of a sequence of perspectivities. It is evident that [cr] and JV ] may be the same form, in which case the projectivity effects a permutation of the elements of the form. For example, it is proved later in this chapter that any four points A, B, C, D of a line can be projected into B, A } D, C respectively. A projectivity establishes a one-to-one correspondence between the elements of two one-dimensional forms, which correspondence we may consider abstractly without direct reference to the sequence of perspec- tivities by which it is defined. Such a correspondence we denote by Projectivities we will, in general, denote by letters of the Greek alphabet, such as TT. If a projectivity TT makes an element a- of a form homologous with an element a 1 of another or the same form, we will sometimes denote this by the relation ^(^)== .';,. Jn this case we may say the* projectivity transforms cr into <r'. Here the symbol TT( ) is used as a functional symbol "f acfiEg orTthe variable J cr, which represents any one of the elements of a given form. * This is Poncelet's definition of a projectivity. t Just like F(x), sin(a;), log(fc), etc. J The definition of variable is " a symbol x which represents any one of a class of elements [cc]." It is in this sense that we speak of " a variable point." 23] PEOJECTIYITY 59 23. The projectivity of one-dimensional primitive forms. The projectivity of one-dimensional primitive forms will be discussed with reference to the projectivity of pencils of points. The corre- sponding properties for the other one-dimensional primitive forms will then follow immediately by the theorems of duality (Theorems 11-13, Chap. I). THEOREM 1. If A, B } C are three points of a line I and A 1 , B r , C f three points of another line V 9 then A can "be projected into A r , B into B f , and C into C 1 ly means of two centers of perspectivity. (The lines may be in the same or in different planes.) (A, E) Proof. If the points in any one of the pairs AA\ BE 1 , or CC' are coincident, one center is sufficient, viz., the intersection of the lines determined by the other two pairs. If each of these pairs consists of distinct points, let 8 be any point of the line AA r , distinct from A and A 1 (fig. 22). From 8 project A, B, C on any line V r distinct from I and V 9 but con- taining A f and a point of I If B", C" are the points of Z" correspond- ing to B, C respectively, the point of intersection S 1 of the lines B f B rf and C f O fr is the second center of perspectivity. This argument holds without modification, if one of the points A 9 B, C coincides with one of the points A r , B ! , O r other than its corresponding point. COROLLARY 1. IfA,B,C and A!, B ! 9 C 1 are on the same line, three centers of perspectivity are sufficient to project A, B, C into A 1 , B f , C f respectively. (A, E) COROLLARY 2. Any three distinct elements of a one-dimensional primitive form are protective with any three distinct elements of another or the same one-dimensional primitive form. (A, E) For, when the two forms are of the same kind, the result is ob- tained from the theorem and the first corollary directly from the 60 PRIMITIVE GEOMETEIO FOKMS [CHAP.III theorems of duality (Theorems 1113, Chap. I). If they are of differ- ent kinds, a projection or section is sufficient to reduce them to the same kind. THEOREM 2. The projectivity ABCD -j^B ADC holds for any four distinct $>oints A, B, C, D of a line. (A, E) Proof. From a point S, not on the line I = AB, project ABCD into AB r C f D f on a line V through A and distinct from I (fig. 23). From D project AB'C f D f on the line SB. The last four points will then project into BADC by means of the center C f . In fig. 23 we have S D C r ABCD~=AB r C f D r = BB f C"S = BADC. A A A It is to be noted that a geometrical order of the points ABCD has no bearing on the theorem. In fact, the notion of such order has not yet been introduced into our geometry and, indeed, cannot be introduced on the basis of the present assumptions alone. The theo- rem merely states that the correspond- ence obtained ly interchanging any two of four collinear points and also inter- changing the remaining two is projectwe. The notion of order is, however, im- plied in our notation of projectivity and perspectivity. Thus, for example, we introduce the following definition : DEFINITION. Two ordered pairs of elements of any one-dimensional form are called a throw;"]! the pairs are AB, CD, this is denoted by T(AB, CD). Two throws are said to be equal, provided they are projective ; in symbols, T(AB, CD) = T (A f B f , C'D f ), provided we have The last theorem then states the equality of throws : J(AB, CD) = T(BA, DC) = T(CD, AB) = T(DC, The results of the last two theorems may be stated in the follow- ing form : THEOREM 1'. If 1 , %, 3 are elements of any one-dimensional prim- itive form, there exist projective transformations 'which will effect any one of the six permutations of these three elements. PEOJECTIYITY 61 THEOREM 2'. If 1, 2, 3, 4 are any four distinct elements of a one- dimensional primitive form, there exist ^rojectixe transformations which will transform 1234 into an y ne f ^ ie following permuta- tions of itself: 1234, 143, 3412, 4321. A projective transformation has been defined as the resultant of any sequence of perspectivities. We proceed now to the proof of a chain of theorems, "which lead to the fundamental result that any projective transformation between two distinct one-dimensional primitive forms of the same kind can be obtained as the resultant of two perspectivities. THEOREM 3. If [P], [P'], [P' f ] are pencils of points on three distinct S S 1 concurrent lines I, V, l n respectively, sucli that [P] = [P f ] and [P r ] = S" "], then likewise [P] = [P"], and the three centers of perspectivity S, S f S" are collinear. (A, E) TIG-. 24 Proof. Let be the common point of the lines 1 9 V, l n . 1S.P V P^P Z are three points of [P], and ^J^'J?' and JJ /7 J5"JJ f/ the corresponding points of [P'], [P n ] (fig. 24), it is clear that the triangles P^P^, P^PI, P^P^ !I are perspective from 0* By Desa-jg^e^Jbh^eorgm (Theorem 1, Chap. II) homologous sides of ^any pair, of these three triangles meet in ^nll jnaar ? ppiTrf^ Thp finn elusion of the theorem then follows readily from the hypotheses. * If the points in each of these sets of three are collinear, the theorem is obvious and the three centers of perspectivity coincide. 62 PRIMITIVE GEOMETEIC FOEMS [CHAP. Hi COROLLARY. If n concurrent lines l v l v Z 8 , S, n-l,n (A,E) , l n are connected by if I perspectivities L ^_, A L - 2J A L - aj A are distinct lines, then we have [J^] == | Proof. This follows almost immediately from the theorem, except when it happens that a set of four successive lines of the set Z 1 Z 2 Z S - l n are such that the first and third coincide and likewise the second and fourth. That this case forms no exception to the corollary may be shown as follows : Consider the perspectivities connecting the pencils of points on the lines l v Z 2 , Z 3 , Z 4 on the hypothesis that Z 1 = Z 8 , Z a = Z 4 (fig. 25.) Let l v l z meet in <9, and let the line 3^S M meet Z x in A v and l z in -4 2 ; let A 3 = A l and J^ 4 be the corresponding points of Z 3 and Z 4 respectively. Further, let JS V B Z9 B and C v C z> <7 8 , (7 4 be any other two sequences of corresponding points in the perspectivities. Let S A be determined as the intersection of the lines A^A.^ and B^B^ The two quadrangles S 12 S^B 2 C 2 and $ 41 34 J? 4 <7 4 have five pairs of homologous sides meeting l t =l s in the points OA^B^B^Cy /Hence the side S^C^ meets ^ in C^ (Theorem 3, Chap. II). THEOREM 4. J/ K] K] [^] ^ re pencils of points on distinct cr Sf Kwe^ Z y Z 2 , Z respectively, such that K] = [^]==K], awd ^ [P'] is the pencil of points on any line V containing the intersection of l v I and also a point of 1 2 , "but not containing $ 2 , then there exists a point S f S v such that [JJ] = [P ! ] = [J>]. (A, E) on 23] PEOJEGTIYITY 63 Proof. Clearly we have But by tlie preceding theorem and the conditions on V we have # [-ZJ] = [P'], -where S a ' is a point of ^^ Hence we have K] = [^'] = K]- This theorem leads readily to the next theorem, which is the result toward which we have been working. We prove first the following lemmas : LEMMA 1. Any axial perspectivity between the points of two skeiv lines is equivalent to (and may le replaced ly) two central perspectives. (A,E) For let [P], [P'] be the pencils of points on the skew lines. Then if S and S r are any two points on the axis s of the axial perspectivity, the pencils of lines $[P], S r [P f ] * are so related that pairs of homol- ogous lines intersect in points of the line common to the planes of the two pencils [P] and S f [P f ], since each pair of homologous lines lie, by hypothesis, in a plane of the axial pencil s[P]~s[P f ]. LEMMA 2. Any projectivity "between pencils of points may be defined "by a sequence of central perspectivities. For any noncentral perspectivities occurring in the sequence defining a projectivity may, in consequence of Lemma 1, be replaced by sequences of central perspectivities. THEOREM 5. If two pencils of points [P] and [P ; ] on distinct lines are projective, there exists a pencil of points [Q] and two points S, S r S S f such that we haw [P] = [Q] = [P']. (A, E) Proof. By hypothesis and the two preceding lemmas we have a sequence of perspectivities & & S S S n * Given a class of elements [P]; the symbol S[P] is used to denote the class of elements SP determined by a given element S and any element of [P]. Hence, if [P] is a pencil of points and 8 a point not in [P], S [P] is a pencil of lines with center 8 ; if s is a line not on any P, s [P] is a pencil of planes with axis s. 64 PEIMITIVE GEOMETEIC POEMS [CHAP, in We assume the number of these perspectivities to be greater than two, since otherwise the theorem is proved. By applying the corollary of Theorem 3, when necessary, this sequence of perspectivities may be so modified that no three successive axes are concurrent. We may also assume that no two of the axes Z, Z p Z 2 , Z s , , V of the pencils [P], [J], [JJ], [JJ], - [P'J are coincident; for Theorem 4 may evidently be used to replace any l t (= J t ) by a line ZJf (= Z t ). Now let l[ be the line joining the points ZZ t and Z 2 Z 8 , and let us suppose that it does not contain the center 2 (fig. 26). If then [2J 7 ] is the pencil of points on l[ t we may (by Theorem 4) replace the given sequence of per- orf or or cr spectivities by [P] == [P^] = [P 2 ] = [P s ] = and this sequence may in turn be replaced by t sy si s, (Theorem 3). If $ 2 is on the line joining ll : and Z 2 Z 3 , we may replace Z x by any line I" through the inter- l section of Z X Z 2 which meets Z and y IG t 26 does nofc contain the point 3 l (The- orem 4). The line joining Z 2 Z 3 to ZZ[' does not contain the point 3% which replaces 3 S . For, since ^ 2 is on the line joining Z 3 Z 2 to ll v the points Z 3 Z 2 and ZZ X are homologous points of the pencils [P 8 ] and [P] ; and if 3% were on the line join- ing Z 3 Z 2 to ll[ r , the point Z 3 Z 2 would also be homologous to ll[ f . We may then proceed as before. By repeated application of this process we can reduce the number of perspectivities one by one, until finally we obtain the pencil of points [Q] and the perspectivities S S r ' As a consequence we have the important theorem : THEOEEM 6. Any two projective pencils of points on skew lines are axially perspective. (A, E) Proof. The axis of the perspectivity is the line 33 r of the last theorem. 24. General theory of correspondence. Symbolic treatment. In preparation for a more detailed study of projective (and other) corre- spondences, we will now develop certain general ideas applicable to CORRESPONDENCE 65 all one-to-one reciprocal correspondences as defined in Chap. II, 13, p. 35, and show in particular how these ideas may be conveniently represented in symbolic form.* As previously indicated (p. 58), we will represent such correspondences in general by the letters of the Greek alphabet, as A, B, F, . The totality of elements affected by the correspondences under consideration forms a system which we may denote by S. If, as a result of replacing every element of a system $! by the element homologous to it in a correspondence A, the sys- tem Sj is transformed into a system S 2 , we express this by the relation A(S 1 ) = S a . In particular, the element homologous with a given ele- ment P is represented by A (P). I. If two correspondences A, B are applied successively to a sys- tem Sj, so that we have A (S t ) = S 2 and B (S 2 ) = S 3 , the single corre- spondence F which transforms S x into S 3 is called the resultant or product of A by B; in symbols S s = B (S 2 ) = B (A(S 1 )) = BA (SJ, or, more briefly, B A = F. Similarly, for a succession of more than two correspondences. II. Two successions of correspondences A m A m ^i - A x and BgB^-L - B! have the same resultant, or their products are equal, provided they transform S into the same S'; in symbols, from the A W A W _ 1 - - A,(S) = B A.! - - - B,(S) follows A^A^ - . - A x = B ff B^. . . B r III. The correspondence which makes every element of the sys- tem correspond to itself is called the identical correspondence or simply the identity, and is denoted by the symbol 1. It is then readily seen that for any correspondence A we have the relations IV. If a correspondence A transforms a system S x into S 2 , the corre- spondence which transforms S 3 into S x is called the inverse of A and is represented by A" 1 ; i.e. if we have A (S x ) = S 2 , then also A"" 1 (S 2 ) = S r The inverse of the inverse of A is then clearly A, and we evidently have also the relations AA- I =AT I A=I. * In this section we have followed to a considerable extent the treatment given "by H. Wiener, Berichte der K. sachsischen Gesellschaft der Wissenschaften, Leipzig. Vol. XLH (1890), pp. 249-252. 66 PEIMITIYE GEOMETBIC FOBMS [CHAP. Ill Conversely, if A, A' are two correspondences such that we have AA' = 1, then A' is the inverse of A. Evidently the identity is its own inverse. V. The product of three correspondences A, B, F always satisfies the relation (FB) A = F (BA) (the associative law). For from the relations A(S 1 )=S 2 , B(S 2 )=S 3 , F(S 3 )=S 4 foUows at once BA(S 1 )=S 3 , whence F(BA) (S~) =S 4 ;" and also FB (S 2 ) = S 4 , and hence (FB) A (SJ = S 4 , which proves the relation in question. More generally, in any product of correspondences any set of successive correspondences may be inclosed in parentheses (provided their order be left unchanged), or any pair of parentheses may be removed; in other words, in a product of correspondences any set of successive correspondences may be replaced by their resultant, or any correspondence may be replaced by a succession of which the given correspondence is the resultant. VI. In particular, we may conclude from the above that the inverse of the product M - - BA is A" 1 B"" 1 M" 1 , since we evidently have the relation M - - - BAA^B" 1 . . . M"^! (cf. IV). VII. Further, it is easy to show that from two relations A = B and F = A follows AF = BA and FA = AB. In particular, the relation A = B may also be written AB~ x = 1, B~ x A = 1, B A~ l = 1, or A~ *B = 1. VIII. Two correspondences A and B are said to be commutative if they satisfy the relation BA = AB. IX. If a correspondence A is repeated n times, the resultant is writ- ten AA A = A n . A correspondence A is said to be of period n, if n is the smallest positive integer for which the relation A n = 1 is satisfied. When no such integer exists, the correspondence has no period ; when it does exist, the correspondence is said to be periodic or cyclic. X. The case n = 2 is of particular importance. A correspondence of period two is called involutoric or reflexive. 25. The notion of a group. At this point it seems desirable to introduce the notion of a group of correspondences, which is funda- mental in any system of geometry. We will give the general abstract definition of a group as follows : * DEFINITION. A class G of elements, which we denote by a, &, c, -, is said to form a group with respect to an operation or law of * We have used here substantially the definition of a group given by L. E. Dickson, Definitions of a Group and a Pield by Independent Postulates, Transactions of the American Mathematical Society, Vol. VI (1905), p. 199. 25,26] GROUPS 67 combination o, acting on pairs of elements of G, provided the fol- lowing postulates are satisfied : G 1, For every pair of (equal or distinct) elements a, b of G, the result a o & of acting with the operation o on the pair in the order given * is a uniquely determined element of G. G 2. The relation (a o &) o c = a o (b o c) holds for any three (equal or distinct] elements a, 1), c of G. G3. There occurs in G an element i, such that the relation aoi = a holds for ever?/ clement a of G. G4. For every element a in G there exists an element a 1 satisfying the relation a o a'= i. From the above set of postulates follow, as theorems, the following : The relations a o a r = i and a o i = a im%)ly respectively the relations a'o a i and ioa~ a. An element i of G is called an identity element, and an element a 1 satisfying the relation a o a 1 = i is called an inrerse element of a. There is only one identity element in G. For every element a of & there is only one inverse. We omit the proofs of these theorems. DEFINITION. A group which satisfies further the following postulate is said to be commutative (or dbelian) : G 5. TJie relation a o 5 = I o a is satisfied for every pair of de- ments a, b in G. 26. Groups of correspondences. Invariant elements and figures. The developments of the last two sections lead now immediately to the theorem: A set of correspondences forms a group provided the set contains the inverse of any correspondence in the set and provided the resultant of any two correspondences is in the set Here the law of combination o of the preceding section is simply the formation of the resultant of two successive correspondences. DEFINITION. If a correspondence A transforms every element of a given figure F into an element of the same figure, the figure F is said to be invariant under A, or to be left invariant by A. In particular, * I.e. ao 5 and & o a are not necessarily identical. The operation o simply defines a correspondence, whereby to every pair of elements a, 6 in G in a given order corre- sponds a unique element j this element is denoted by a o &. 68 PEIMITIVE G-EOMETEIC FOEMS [CHAP, in an element which is transformed into itself by A is said to be an invariant element of A; the latter is also sometimes called a double element or a fixed element (point, line, plane, etc.). We now call attention to the following general principle : The set of all correspondences in a group G which learn a given figure invariant forms a group. This follows at once from the fact that if each of two corre- spondences of G leaves the figure invariant, their product and their inverses will likewise leave it invariant ; and these are all in G, since, by hypothesis, G is a group. It may happen, of course, that a group defined in this way consists of the identity only. These notions are illustrated in the following section : 27. Group properties of projectivities. From the definition of a pro- jectivity between one-dimensional forms follows at once THEOREM 7. The inverse of any projectivity and the resultant of any two projectivities are projectivities. On the other hand, we notice that the resultant of two perspec- tivities is not, in general, a perspectivity ; if, however, two perspec- tivities connect three concurrent lines, as in Theorem 3, their resultant is a perspectivity. A perspectivity is its own inverse, and is therefore reflexive. As an example of the general principle of 26, we have the important result : THEOREM 8. The set of all projectivities leaving a given pencil of points invariant form a group. If the number of points in such a pencil is unlimited, this group con- tains an unlimited number of projectivities. It is called the general projective group on the line. Likewise, the set of all projectivities on a line leaving the figure formed by three distinct points invariant forms a subgroup of the general group on the line. If we assume that each per- mutation (cf. Theorem I/) of the three points gives rise to only a single projectivity (the proof of which requires an additional assumption), this subgroup consists of six projectivities (including, of course, the identity). Again, the set of all projectivities on a line leaving each of two given distinct points invariant forms a subgroup of the general group. We will close this section with two examples illustrative of the principles now under discussion, in which the projectivities in ques- tion are given by explicit constructions. 27] G-BOUP OF PEOJECTIVITIES 69 EXAMPLE 1. A group of projectivities leaving eacli of two given points invariant. Let M 9 N be two distinct points on a line I, and let m, n be any two lines through 2I 9 N respectively and coplanar with I (fig. 27). On m let there be an arbitrary given point S. If ^ is any other point on m distinct from J/ ; the points 8, S l together with the line n define a projectivity TT^ on I as follows : The point TTj (A) = A r homologous to any point A of I is obtained by the two S 8 l perspectivities [A] = [^i]~[-4/], where [A^ is the pencil of points on n. Every point jS % then, distinct from J/, defines a unique pro- jectivity 7r 4 ; we are to show that the set of all these projectivities TT, forms a group. We show first that the product of any two TT V 7r 2 is a uniquely determined pro- jectivity 7T 3 of the set (fig. 27). In the figure, A! = TT^ (A) and A n = Tr z (A f ) have been M B B' B" A A' A" N constructed. The point S s giving A n directly from A by a similar con- struction is then uniquely determined as the intersection of the lines A n A v m. Let B be any other point of I distinct from M 9 N 9 and let J5'= ir^(S) and J?*= ^(B 1 ) be constructed ; we must show that we have B"=z*jr z (B). We recognize the quadrangular set Q(If3'A r , NA"B n ) as defined by the quadrangle SS 2 B Z A Q . But of this quadrangular set all points except B n are also obtained from the quadrangle S^B^A^ whence the line S^ determines the point B n (Theorem 3, Chap. II). It is necessary further to show that the inverse of any projectivity in the set is in the set. For this purpose we need simply determine $ 2 as the intersection of the line AA Z with m and repeat the former argu- ment. This is left as an exercise. Finally, the identity is in the set, since it is TT V when S^S. 70 PRIMITIVE GEOMETRIC FORMS [CHAP.III It is to be noted that in this example the points Jf and N are double points of each projectivity in the group; and also that if P, P r and Q, Q r are any two pairs of homologous points of a projectivity we have Q (MPQ, NQ'P ! ). Moreover, it is clear that any projectivity, of the group is uniquely determined by a pair of homologous elements, and that there exists a projectivity which will transform any point A of I into any other point B of I, provided only that A and B are distinct from M and N. By virtue of the latter property the group is said to be transitive. A A' A/ A" FIG. 28 EXAMPLE 2. Commutative projectivities. Let M be a point of a line Z, and let m, m f be any two lines through H distinct from I, but in the same plane with I (fig. 28.) Let 8 be a given point of m } and let a projectivity TT I be defined by another point S l of m which deter- *? a mines the perspectivities [A] = [-4 X ] == [A r ] 9 where [^ x ] is the pencil of points on m f . Any two projectivities defined in this way ly points S l are commutative. Let 7r 2 be another such projectivity, and construct the points A r =^(A) 9 A' f ='jr 2 (A r ) 9 and A[ = Tr z (A). The quadrangle JSS^A^A^ gives Q(MAA r , ITA ff A[)- and the quadrangular set determined on I by the quadrangle SS^A^ has the first five points of the former in the same positions in the symbols. Hence we have vr^A^) = A rr , and therefore TT^ = TT^. EXERCISES 1. Show that the set of all projectivities TT^ of Example 2 above forms a group, which is then a commutative group. 2. Show that the projectivity wj of Example 1 above is identical with the projectivity obtained by choosing any other two points of m as centers of perspectivity, provided only that the two projectivities have one homologous 27,28] TWO-DBmNSIOjSTAL PEOJEOTIYITIES 71 pair (distinct from M or 2V) in common. Investigate the general question as to how far the construction may be modified so as still to preserve the propo- sition that the projectivities are determined by the double points J/, -V and one pair of homologous elements. 3. Discuss the same general question for the projectivities of Example 2. 4. Apply the method of Example 2 to the projectivities of Example 1. "Why does it fail to show that any two of the latter are commutative ? State the space and plane duals of the two examples. 5. ABCD is a tetrahedron and a, ^8, y, S the faces not containing A ,B,C,D respectively, and I is any line not meeting an edge. The planes (I A , IB, 1C, ID) are projective with the points (la, 1/3, ly, /S). 6. On each of the ten sides of a complete 5-point in a plane there are three diagonal points and two vertices. Write down the projectivities among these ten sets of five points each. 28. Projective transformations of two-dimensional forms. DEFINITION. A projective transformation between the elements of two two-dimensional or two three-dimensional forms is any one-to- one reciprocal correspondence between the elements of the two forms, such that to every one-dimensional form of one there corresponds a projective one-dimensional form of the other. DEFINITION. A collineation is any (1, 1) correspondence between -"^j^a^iEWM*--* J \ / JT two two-dimensionaior two three-dimensional forms in which to every element of one of the forms corresponds an element of the same kind in the other form, and in which to every one-dimensional form of one corresponds a one-dimensional form of the other. A. projective, colline- ation is one in which this correspondence is projective. Unless other- wise specified, the term collineation will, in the future, always denote a projective collineation.* In the present chapter we shall confine ourselves to the discus- sion of some of the fundamental properties of collineations. In this section we discuss the collineations between two-dimensional forms, and shall take the plane (planar field) as typical; the corresponding theorems for the other two-dimensional forms will then follow from duality. The simplest correspondence between the elements of two distinct pla r nes TT, ir 1 is a perspective correspondence, whereby any two homol- ogous elements are on the same element of a bundle whose center is on neither of the planes TT, TT'. The simplest collineation in a plane, * In how far a collineation must be projective will appear later. 72 PRIMITIVE GEOMETEIC FOEMS [CHAP.III ie. which transforms every element of a plane into an element of the same plane, is the following : DEFINITION. A perspectwe^ collineation in a plane is a protective collineation leaving invariant every point on a given line o and every, line on a given point 0. The line o and the point are called the axis and center respectively of the perspective collineation. If the center and axis are not united, the collineation is called a planar homology; if they are united, a planar elation. A. perspective collineation in a plane TT may be constructed as follows : Let any line o and any point of TT "be chosen as axis and center respectively, and let T^ be any plane through o distinct from TT. Let O v 2 be any two points collinear with and in neither of the planes TT, 7r r The perspective collineation is then obtained by the O l 0* two perspectivities [P] === [PJ = [P'], where P is any point of TT and .ZJ, P r are points of TT I and TT respectively. Every point of the line o and every line through the point clearly remain fixed by the trans- formation, so that the conditions of the definition are satisfied, if only the transformation is projeetive. But it is readily seen that every pencil of points is transformed by this process into a perspec- tive pencil of points, the center of perspectivity being the point 0\ and every pencil of lines is transformed into a perspective pencil, the axis of perspectivity being o. The above discussion applies whether or not the point is on the line o, THEOREM 9. A perspective col- lineation in a plane is uniquely defined if the center, axis, and any two homologous points (not on the axis or center) are given, with the single restriction that the homol- ogous points must le collinear with 0. (A, E) Proof. Let 0, o be the center and axis respectively (fig. 29). It is clear from the definition that any two homologous points must be collinear with 0, since every line through. is invariant ; similarly (dually) any two homologous lines must be concurrent with o. Let A, A 1 be the given pair of homologous points collinear with 0. The 28] TWO-DIMEXSIO-X T AL PEOJECTITITIES 73 point B f homologous to any point B of the plane is then determined. We may assume 3 to be distinct from 0, A, A 1 and not to be on o. B ! is on the line OB, and if the line AB meets o in C, then, since C is invariant by definition, the line AB = AC is transformed into A r C. B f is then determined as the intersection of the lines OB and A'C. This applies unless B is on the line A A'} in this case we determine as above a pair of homologous points not on AA' 9 and then use the two points thus determined to construct B 1 . This shows that there can be no more than one perspective collineation in the plane with the given elements. To show that there is one we may proceed as follows : Let ^ be any plane through o distinct from TT, the plane of the perspectivity, and let O t be any point on neither of the planes TT, TT I . If the line A0 1 meets ^ in A v the line A ? A^ meets 00^ in a point 2 . The perspec- tive collineation determined by the two centers of perspectivity O v and the plane ^ then has 0, o as center and axis respectively and A 9 A r as a pair of homologous points. COROLLARY 1. A perspective collineation in a plane transforms every one-dimensional form into a perspective one-dimensional form. (A, E) COROLLARY 2. A perspective collineation with center and axis o transforms any triangle none of whose vertices or sides are on o *or into a perspective triangle, the center of perspectivity of the triangles being the center of the collineation and the asois of perspectivity being the axis of the collineation. (A, E) COROLLARY 3. The only planar collineations (whether required to be protective or not] which leave invariant the points of a line o and the lines through a point are homologies if is not on o, and elations if is on o. ,(A, E) Proof. This will be evident on observing that in the first paragraph of the proof of the theorem no use is made of the hypothesis that the collineation is protective. COROLLARY 4. If H is a perspective collineation such that H (0) = 0, H(o) = o, E(A) A r 9 H(B) = B f where A, A r , B, B J are collinear with a point K of o, then we have Q(OAB, KB'A 1 ). (A, E) Proof. If C is any point not on AJ! and H(<7) = C", the lines AC and A r C r meet in a point L of o, and BO and B f C r meet in a point M of o\ and the required quadrangle is CC'LM (cf. fig. 32, p. 77). 74 PRIMITIVE GEOMETRIC FORMS [CHAP, m THEOREM 10. Any complete quadrangle of a plane can le trans- formed into any complete quadrangle of the same or a different plane ly a projectile collineation which, if the quadrangles are in the same plane, is the resultant of a finite numler of perspective collineations. (A,E) Proof. Let the quadrangles be in the same plane and let their ver- tices be A, B, 0, D and A', B\ C", D J respectively. We show first that there exists a collineation leaving any three vertices, say A', B 1 , C f , of , yl / / v >\/Vi \ ' k / / / / A' T 0* Pio. 80 the quadrangle A f B ! C f D r invariant and transforming into the fourth, D f , any other point Z> 8 not on a side of the triangle A r $ r C r (Gg- 30). Let D be the intersection of ^D 3 , ffD 1 and consider the homology with center A 1 and ajds JS f C f transforming D 3 into D. Next consider the homology with center B 1 and axis C f A r transforming ^into D f . Both these homologies exist by Theorem 9. The resultant of these two homologies is a eollineation leaving fixed A 1 , B r , C 1 and transforming Z> 3 into D f . (It should be noticed that one or both of the homologies may be the identity.) Let O t be any point on the line containing A and A r and let o be any line not passing through A or A*. By Theorem 9 there exists a 28,29] THEBB-DIMEirSIOSrAL PBOJECTIVITIES 75 perspective collineation ^ transforming A to A* and having O l and t as center and axis. Let B v C v D^ be points such that In like manner, let o z be any line through A f not containing B^ or B r and let 2 be any point on the line B^B'. Let 7r 3 be the perspec- tive collineation with axis 2 , center 2 , and transforming B : to B*. Let <7 = 7 and D= w). Here" 3 Now let 3 be any point on the line C 2 6 f/ and let 7r 8 be the per- spective collineation which has A f B r =o s for axis, 3 for center, and transforms <7 2 to <7 ; . The existence of 7r g follows from Theorem 9 as soon as we observe that C f is not on the line A f B f , by hypothesis, and 6 2 is not on A ! B ! \ because if so, C^ would be on A r B and there- fore C would' be on AB. Let 7r 3 (D 2 ) = D 3 . It follows that The point D 3 cannot be on a side of the triangle A f B f C f because then D 2 would be on a side t of A ! B r C 2 , and hence D^ on a side of A r B : G v and, finally, D on a side of -ABC. Hence, by the first para- graph of this proof, there exists a projectivity 7r 4 such that n-t(A r B r C'Z> 9 ) = A'B'C'D*. The resultant tr^ir^ir^rr^ of these four collineations clearly transforms A, B, C, D into A f 9 B r , C f , D ! respectively. If the quadrangles are in different planes, we need only add a perspective transformation between the two planes. COROLLAEY. There exist projectile collineations in a plane which will effect any one of the possible % permutations of the vertices of a complete quadrangle in the plane. (A, E) 29. Projective collineations of three-dimensional forms. Protective collineations in a three-dimensional form have been defined at the beginning of 28. DEFINITION. A projective collineation in space which leaves inva- riant every point of a plane a> and every plane on a point is called a perspective collineation. The plane a> is called the plane ofperspectivity; the point is called the center. If is on a>, the collineation is said to be an elation in space ; otherwise, a homology in space. 76 PEIMITIYE GBOMETEIC FOKMS [CHAP.III THEOREM 11. I/O is any point and a any plane, there exists one and only one perspective collineation in space having 0, co for center and plane of per spectivity respectively, which transforms any point A (distinct from and not on co) into any other point A! (distinct from and not on co) collinear with AO. (A, E) Proof. We show first that there cannot be more than one per- spective collineation satisfying the conditions of the theorem, by showing that the point B f homologous to any point B is uniquely determined by the given conditions. We may assume B not on a and distinct from and A. Suppose first that B is not on the line AO (fig. 31). Since BO is an invariant line, B 1 is on B0\ and if the line AB meets co in L, the line AB = AL is transformed into the line A ! L. Hence B r is determined as the intersection of BO and A r L. There remains the case where B is on AO and distinct from A and (fig. 32). Let 0, C 1 be any pair of homologous points not on AO, and let AC and BC meet to in L and M respectively. The line MB = M is transformed into M C r 9 and the point B J is then determined as the intersection of the lines BO and MC f . That this point is independent of the choice of the pair C, C J now follows from the fact that the quadrangle MLCC 1 gives the quadrangular set Q(KAA f , OB r B), where K is the point in which AO meets co (K may coincide with without affecting the argument). The point B* is then uniquely determined by the five points 0, E, A, A 1 , B. The correspondence defined by the construction in the paragraph above has been proved to be one-to-one throughout. On the line AO it is projective because of the perspectivities (fig. 32) 29] THBEE-DIMENSIO]S T AL PEOJECTIYITIES 77 C ' On OB, any other line through 0, it is projeetive because of the per- spectivities (fig. 31) , ^ w=w= That any pencil of points not through is transformed into a perspective pencil, the center of perspectivity being 0, is now easily seen and is left as an exercise for the reader. From this it follows PIG. 32 that any one-dimensional form is transformed into a projeetive form, so that the correspondence which has been constructed satisfies the definition of a projeetive collineation. THEOREM 12. Any complete five-point in space can le transformed into any other complete Jive-point in space ly a projeetive collineation which is the resultant of a finite number of perspective collineations. (A,E) Proof. Let the five-points be ABODE and AWC'D'E* respectively. We will show first that there exists a collineation leaving A f f C f D f invariant and transforming into JS f any point 2S Q not coplanar with three of the points A f J3 r C f Z> f . Consider a homology having A f B ! C l as plane of perspeetivity and D 1 as center. Any such homology trans- forms J into a point on the line JS Q I> f . Similarly, a homology with plane A f f l> ! and center C r transforms Iff into a point on the line HfC r . If E Q D ! and S r O f intersect in a point JS V the resultant of two homol- ogies of the kind described, of which the first transforms H into JS^ and the second transforms JE^ into Jff, leaves A f ffC f Z> f invariant and transforms JH Q into If. If the lines E^D* and JS f C r are skew, there is a line through B 1 meeting the lines E^D 1 and E f C r respectively 78 PEIMITIVE GEOMETEIC POEMS [CHAP, in in two points E^ and JS 2 . The resultant of the three homologies, of which the first has the plane A*'C f and center D f and transforms JSo to JE V of which the second has the plane A f C f D f and center B 1 and transforms E l to JE Z , and of which the third has the plane A f 'D r and center C r and transforms J 2 to E ! , is a collineation leaving A J 3 f C 1 D t invariant and transforming E^ to E 1 . The remainder of the proof is now entirely analogous to the proof of Theorem 10. The details are left as an exercise. COROLLARY. There exist projectile collineations which, will effect any one of the possible 120 permutations of the vertices of a complete five-point in space. (A, E) ' EXERCISES 1. Prove the existence of perspective collineations in a plane without making use of any points outside the plane. 2. Discuss the figure formed by two triangles which are homologous under an elation. How is this special form of the Desargues configuration obtained as a section of a complete five-point in space? 3. Given an elation in a plane with center and axis o and two homol- ogous pairs A , A' and B, B' on any line through 0, show that we always have Q(OAA', OB'B). 4. What permutations of the vertices of a complete quadrangle leave a given diagonal point invariant ? every diagonal point ? 5. Write down the permutations of the six sides of a complete quadrangle brought about by all possible permutations of the vertices. 6. The set of all homologies (elations) in a plane with the same center and axis form a group. 7. Prove that two elations in a plane having a common axis and center are commutative. Will this method apply to prove that two homologies with common axis and center are commutative? 8. Prove that two elations in a plane having a common axis are commu- tative. Dualize. Prove the corresponding theorem in space. 9. Prove that the resultant of two elations having a common axis is an elation. Dualize. Prove the corresponding theorem in space. What groups of elations are defined by these theorems ? 10. Discuss the effect of a perspective collineation of space on : (1) a pencil of lines ; (2) any plane ; (3) any bundle of lines j (4) a tetrahedron ; (5) a complete five-point in space. 11. The set of all collineations in space (in a plane) form a group. 12. The set of all projective collineations in space (in a plane) form a group. 13. Show that under certain conditions the configuration of two perspective tetrahedra is left invariant by 120 collineations (cf. Ex. 3, p. 47). CHAPTER IV HARMOITCC CONSTRUCTIONS AM) THE FUNDAMENTAL THEOREM OF PROTECTIVE GEOMETRY 30. The projectivity of quadrangular sets. We return now to a more detailed discussion of the notion of quadrangular sets introduced at the end of Chap. II. We there defined a quadrangular set of points as the section by a transversal of the sides of a complete quadrangle ; the plane dual of this figure we call a quadrangular set of lines;* it consists of the projection of the vertices of a complete quadrilateral from a point which is in the plane of the quadrilateral, but not on any of its sides ; the space dual of a quadrangular set of points we call a quadrangular set of planes; it is the figure formed by the projection from a point of the figure of a quadrangular set of lines. We may now prove the following im- portant theorem : THEOREM 1. TJie section by a transversal of a quadrangular set of lines is a quadrangular set of points. (A,E) Proof. By Theorem 3', Chap. II, p. 49, and the dual of Note 2, on p. 48, we may take the transversal I to be one of the sides of a com- plete quadrilateral the projection of whose vertices from a point P forms the set of lines in question (fig. 33). Let the remaining three sides of such a quadrilateral be a, &, <?. Let the points Jo> ca, and al * It would be more natural at this stage to call such a set a quadrilateral set of lines; the next theorem, however, justifies the term we have chosen, which has the advantage of uniformity. 79 FIG. 33 80 THE FUNDAMENTAL THEOEEM [CHAP. IV be denoted by A, B, and C respectively. The sides of the quadrangle PABCra&et I in the same points as the lines of the quadrangular set of lines. COROLLARY. A set of collinear points which is projective with a quadrangular set is a quadrangular set. (A, E) THEOKEM I/. The projection from a point of a quadrangular set of points is a quadrangular set of lines. (A, E) This is the plane dual of the preceding ; the space dual is : THEOREM I/ 7 . The section ly a plane of a quadrangular set of planes is a quadrangular set of lines. (A, E) COROLLARY. If a set of elements of a primitive one-dimensional form is protective with a quadrangular set, it is itself a quadrangular set. (A,E) 31. Harmonic sets. DEFINITION. A quadrangular set Q (123, 124) is called a harmonic set and is denoted by H (12, 34). The elements 3, 4 are called harmonic conjugates with respect to the elements 1, 2 ; and 3 (or 4) is called the harmonic conjugate of 4 (or 3) with respect to 1 and 2. From this definition we see that in a harmonic set of points \\(AC, BD), the points A and C are diagonal points of a complete FIG. 34 quadrangle^ while the points B and D are the intersections of the remaining two opposite sides of the quadrangle with the line AC (fig. 34). Likewise, in a harmonic set of lines H (ac, Id), the lines a and c are two diagonal lines of a complete quadrilateral, while the 31] HAEMO]SriC SETS 81 lines 5 and d are the lines joining the remaining pair of opposite vertices of the quadrilateral to the point of intersection ac of the lines a and c (fig. 35). A harmonic set of planes is the space dual of a harmonic set of points, and is therefore the projection from a point of a harmonic set of lines. In case the diagonal points of a complete quadrangle are collinear, any three points of a line form a harmonic set and any point is its own harmonic conjugate with regard to any two points collinear with it. Theorems on har- monic sets are therefore trivial in those spaces for which Assumption H is not true. We shall therefore base our reasoning, in this and the following two sections, on Assumption H Q J though most of the theorems are obviously true also in case H is false. This is why some of the theorems are labeled as dependent on Assumptions A and E, whereas the proofs given involve H Q also. The corollary of Theorem 3, Chap. II, when applied to harmonic sets yields the following: THEOREM 2. The harmbnic conjugate of an element with respect to two oilier elements of a one-dimensional primitive form is a unique element of the form. (A, E) Theorem 1 applied to the special case of harmonic sets gives THEOREM 3. Any section or projection of a harmonic set is a harmonic set. (A, E) COROLLARY. If a set of four elements of any one-dimensional prim- itive form is projectile with a harmonic set, it is itself a harmonic set. (A,E) THEOREM 4. If 1 and 2 are harmonic conjugates with respect to 3 and 4, 3 and 4 are harmonic conjugates with respect to 1 and 2. (A, E, H ) Proof. By Theorem 2, Chap. Ill, there exists a projectivity 1234^-3412. But by hypothesis we have H(34, 12). Hence by the corollary of Theorem 3 we have H (12, 34). By virtue of this theorem the pairs 1, 2 and 3, 4 in the expression H (12, 34) play the same r81e and may be interchanged,* *The corresponding theorem for the more general expression Q(12S, 456) cannot be derived without the use of an additional assumption (cf . Theorem 24, Chap. IV). 82 THE FUNDAMENTAL THEOBEM [CHAP.IV THEOREMS. Given two harmonic sets H(12, 34) and H(1'2', 3'4'), there exists a projectivity such that 1234^ 1'2'3'4'. (A, E) Proof. Any projectivity 123 ^ 1'2'3' (Theorem 1, Chap. Ill) must transform 4 into 4' by "virtue of Theorem 3, Cor., and the fact that the harmonic conjugate of 3 with respect to 1 and 2 is unique (Theo- rem 2). This is the converse of Theorem 3, Cor. COROLLABY 1. If H (12, 34) and H (12', 3'4') are two harmonic sets of different one-dimensional forms having the element 1 in common, we ham 1234 = 12'3'4'. (A, E) For under the hypotheses of the corollary the projectivity 123-^ 1'2'3' of the preceding proof may be replaced by the perspectivity 123 == 12'3'. COEOLLAEY 2. If H (12, 3 4) is a harmonic set, there exists a projec- tivity 1234^-1243. (A,E) This follows directly from the last theorem and the evident fact that if H (12, 34) we have also H (12, 43). The converse of this corollary is likewise valid ; the proof, however, is given later in this chapter (ef. Theorem .27, Cor. 5). We see as a result of the last corollary and Theorem 2, Chap. Ill, that if we have H(12, 34), there exist projectivities which will trans- form 1234 into any one of the eight permutations 1234, 1243, 2134, 2143, 3412, 3421, 4312, 4321.* In other words, if we have H(12, 34), we have likewise H(12, 43), H(21, 34), H(21, 43), H(34, 12), H(34, 21), H(43, 12), H(43, 21). THEOREM 6. The two sides of a complete quadrangle which meet in a diagonal point are harmonic conjtyates with respect to the two sides of the diagonal triangle which meet in this point. (A, E) Proof. The four sides of the complete quadrangle which do not pass through the diagonal point in question form a quadrilateral which defines the set of four lines mentioned as harmonic in the way indicated (fig. 36). It is sometimes convenient to speak of a pair of elements of a form as harmonic with a pair of elements of a form of different ^ kind. For example, we may say that two points are harmonic with two lines in a plane with the points, if the points determine two * These transformations form the so-called eigM-group. 31] HABM02SIC SETS 83 lines through the intersection of the given lines which are harmonic with the latter; or, what is the same thing, if the line joining the points meets the lines in two points harmonic with the given points. With this understanding we may restate the last theorem as follows: The sides of a complete quadrangle which meet in a diagonal point are harmonic with the other two diago- nal points. In like manner, we may say that two points are harmonic with two planes, if the line joining the points meets the planes in a pair of points harmonic with the given points ; and a pair of lines is harmonic with a pair of planes, if FIG. 35 they intersect on the intersection of the two planes, and if they determine with this intersection two planes harmonic with the given planes. EXERCISES 1. Prove Theorem 4 directly from a figure without using Theorem 2, Chap. III. 2. Prove Theorem 5, Cor. 2, directly from a figure. 3. Through a given point in a plane construct a line which passes through the point of intersection of two given lines in the plane, without making use _cjfjhe latter point. 4. A line meets the sides of a triangle ABC in the points A I9 B v C 1? and the harmonic conjugates A%, B 2 , C 3 of these points with respect to the two vertices on the same side are determined, so that we have H(^jB, C^C*), H(BC,A l A 2 )^nd.H(CA,B l B^ Show that A 19 B v C 2 ; ^C^A^C^A^ are collinear; that AA^ BB 2 , CC 2 are concurrent; and that AA 2 , BB V CC X ; AA V BB^ CCtf AA^BBv CC 2 are also concurrent. 5. If each of two sides AB, EC of a triangle ABC meets a pair of opposite edges of a tetrahedron in two points which are harmonic conjugates with respect to .4, B and B, C respectively, the third side CA will meet the third pair of opposite edges in two points which are harmonic conjugates with respect to C, A. 6. A, B, C, D are the vertices of a quadrangle the sides of which meet a given transversal I in the six points P 1? P 2 , P s , P 4 , P 5 , P 6 ; the harmonic conju- gate of each of these points with respect to the two corresponding vertices of the 84 THE FUOTDAMElSrTAL THEOEEM [CHAP. IV quadrangle is constructed and these six points are denoted by P{, P%, P%, P, P$, PQ respectively. The three lines joining the pairs of the latter points which lie on opposite sides of the quadrangle meet in a point P, which is the harmonic conjugate of each of the points in -which these three lines meet I with respect to the pairs of points P' denning the lines. 7. Denning the polar line of a point with respect to a pair of lines as the harmonic conjugate line of the point with regard to the pair of lines, prove that the three polar lines of a point as to the pairs of lines of a triangle form a triangle (called the cogredient triangle) perspective to the given triangle. 8. Show that the polar line denned in Ex. 7 is the same as the polar line denned in Ex. 8, p. 52. 9. Show that any line through a point and meeting two intersecting lines Z, V meets the polar of with respect to I, I' in a point which is the harmonic conjugate of with respect to the points in which the line through meets Z, V. 10. The axis of perspectivity of a triangle and its cogredient triangle is the polar line (cf . p. 4C) of the triangle as to the given pointt 11. If two triangles are perspective, the two polar lines of a point on their axis of perspectivity meet on the axis of perspectivity. 12. If the lines joining corresponding vertices of two n-lines meet in a point, the points of intersection of corresponding sides meet on a line. 13. (Generalization of Exs. 7, 10.) The n polar lines of a point P as to the n (n l)-lines of an n-line in a plane form an ra-line (the cogredient w-line) whose sides meet the corresponding sides of the given n-line in the points of a line p. The line p is called the polar of P as to the ra-line.* 14. (Generalization of Ex. 11.) If two si-lines are perspective, the two polar lines of a point on their axis of perspectivity meet on this axis. 15. Obtain the plane duals of the last- two problems. Generalize them to three- and 7i-dimensional space. These theorems are fundamental for the con- struction of polars of algebraic curves and surfaces of the n-th degree* 32. Kets of rationality on a line, DEFINITION. A point P of a line is said to be "harmonically related to three given distinct points A, B t G of the line, provided P is one of a sequence of points A, B, C, IT 1} j6T 2 , _H" 3 , - of the line, finite in number, such that ^ is the harmonic conju- gate of one of the points A, B, C with respect to the other two, and such that every other point H t is harmonic with three of the set A, B, C, H v HI, - - 3 j5" _ r The class of all points harmonically related to three distinct points A, B, C on a line is called the one-dimensional net of rationality defined by A, B, (7; it is denoted by R(A0). A net of rationality on a line is also called a linear net * This is a definition "by induction of the polar line of a point with respect to an n-line. 32] NETS OF BATIOXALITY 85 THEOREM 7. If A, 3, C, D and A 1 , B f , C f , I) 1 are respectively points of two lines stick that ABCD-^A r B f C r Z> f , and if D is harmonically related to A, 3, C, then D r is harmonically related to A r ,B r , C f . (A, E) This follows directly from the fact that the projectivity of the theo- rem makes the set of points Hj which defines D as harmonically related to A,B, C pro jective with a set of points H( such that every harmonic set of points of the sequence A 9 B, C, H v H^ - -, D is homologous with a harmonic set of the sequence^ 7 , B f , C f , JBJ, H 9 ^ '-,D f (Theorem 3, Cor.). COROLLARY. If a class of points on a line is projectile with a net of rationality on a line, it is itself a net of rationality. THEOREM 8. If K, L, JJ are three distinct points of R (ABC), A, B, C are points of R (KLM). (A, E) Proof. From the projectivity ABCKj^ BAKC follows, by Theorem?, that C is a point of R (ABIC). Hence all points harmonically related to A, B, C are, by definition, harmonically related to A, B, K. Since K is, by hypothesis, in the net R (ABC), the definition also requires that all points of R(ABK] shall be points of R(ABC). Hence the nets R(ABC) and R(ABK) are identical; and so R(ABC) = R(ABK) COROLLARY. A net of rationality -on a line is determined ly any distinct three of its points. THEOREM 9. If all but one of the six (or five, or four) points of a quadrangular set are points of the same net of rationality R, this one point is also a point of R. (A, E) Proof. Let the sides of the quadrangle PQRS (fig. 37) meet the line I as indicated in the points A, A^ B,B^ C } C v so that B = B^\ and suppose that the first five of these are points of a net of rationality R =: We must prove that G : is a point of R. Let the pair of lines ES and PQ meet in B f . We then have S M BCB^A = BQB f P = BAjBiCr Since A is in. R(BCBJ, it follows from this projectivity, in view of Theorem 7, that C l is in R (BA&) == R. DEFINITION. A point P of a line is said to be guadrangularly related to three gwen distinct points A, B, C of the line, provided 86 THE FUNDAMENTAL THEOEEM [CHAP. IV P is one of a sequence of points A, B, C, H v H^ H# of the line, finite in number, such that H^ is the harmonic conjugate of one of the points A, B, with respect to the other two, and such that every other point H t is one of a quadrangular set of which the other five belong to the set A, B, C, & v H v - - -, ^_ r COROLLAET. The class of all points Quadrangularly related to three distinct collinear points A, B, C is R(ABC). (A, E) From the last corollary it is plain that R (ABO) consists of all points that can be constructed from A , .5, C by means of points and lines alone ; that is to say, all points -whose existence can be inferred from Assumptions A, E, H . The existence or nonexistence of further points on the line ABC is unde- termined as yet. The analogous class of points in a plane is the system of all points constructive, by means of points and lines, out of four points A,B,C, D, no three of which are collinear. This class of points is studied by an indirect method in the next section. 33. Nets of rationality in the plane. DEFINITION. A point is said to be rationally related to two noncollinear nets of rationality R v R 2 having a point in common, provided it is the intersection of two lines each of which joins a point of R l to a distinct point of R 2 . A line is said to be rationally related to R and R 2 , provided it joins two points that are rationally related to them. The set of all points and lines rationally related to R v R 2 is called the net of rationality in a plane (or of two dimensions) determined by R v R 2 ; it is also called the planar net defined by R x> R 2 . From this definition it follows directly that all the points of R x and R 2 are points of the planar net defined by R v R 2 . 133] NETS OF EATIONALITY 87 THEOKEM 10. Any line of the planar net R 2 defined "by R v R 2 meets Proof. We prove first that if a line of the planar net R 2 meets R v it meets R 2 . Suppose a line I meets R x in A^ it then contains a second point P of R 2 . By definition, through P pass two lines, each of which joins a point of R 1 to a distinct point of R 2 . If I is one of these lines, the proposition is proved ; if these lines are distinct from Z, let them meet R x and R 2 respectively in the points B v B 2 and JJ, J% (fig. 38). If is the common point of R v R 2 , we then have ,, where -4 2 is the point in which I meets the line of R 2 . Hence A* is a point of R 2 (Theorem 7). Now let I be any line of the net R 3 , and let P, Q be two points of the net and on I (def.). If one of these points is a point of R x or R 2 , the theorem is proved by the case just considered. If not, two lines, each joining a point of R l to a distinct point of R s , pass through P; let them meet R l in A v B It and R 2 in A z , B^ respectively (fig. 38). Let the lines QA^ and QB l meet R 2 in A[ and B^ respectively (first case). Then if I meets the lines of R l and R 2 in Jj* and P% respectively, the quadrangle PQA I B 1 gives rise to the quadrangular set Q(I^A^B Z9 OB^) of which five points are points of R 2 ; hence P z is a point of R 2 (Theorem 9). J? is then a point of R x by the first case of this proof. THEOREM 11. The intersection of any two lines of a planar net is a point of the planar net (A, E) 88 THE FUNDAMENTAL THEOEEM [CHAP. IV Proof. This follows directly from the definition and the last theo- rem, except when one of the lines passes through 0, the point common to the two linear nets R 1? R 2 defining the planar net. In the latter case let the two lines of the planar net be l v 1 2 and suppose 1 2 passes through 0, while ^ meets R 1? R 2 in A v A z respectively (fig. 39). If the point of intersection P of IJ 2 were not a point of the planar net, 1 2 would, by definition, contain a point Q of the planar net, dis- tinct from and P. The lines QA l and QA a would meet R 2 and R 1 in two points 2? 2 and 2^ respec- tively. The point <7 2 in which the line PB^ met the line of R 2 would then be the harmonic conjugate FIG. 39 of JB Q with respect to and A 2 (through the quadrangle <7 2 would therefore be a point of R 2 , and hence P would be a point of the planar net, being the intersection of the lines A^A 2 and B^dy THEOREM 12. TJie points of a planar net R 2 on a line of the planar net form a linear net. (A, E) Proof. Let the planar net be defined by the linear nets R x , R 2 and let I be any line of the planar net. Let P be any point of the planar net not on I or R l or R 2 . The lines joining P to the points of R 2 on I meet R x and R 2 by Theorems 10 and 11. Hence P is the center of a perspectivity which makes the points of R 2 on I perspective with points of R 1 or R 2 . Hence the points of I belonging to the planar net form a linear net. (Theorem 7, Cor.) COROLLARY. The planar net R* defined ly two linear nets R v R 2 is identical with the planar net R| defined "by two linear nets R 8 , R 4 , pro- vided R s , R 4 are linear nets in R^. (A, E) For every point of R 2 is a point of R| by the above theorem, and every point of R^ is a point of Rf by Theorem 10. 33,34] NETS OF KATIONALITY 89 EXERCISE If A, B 9 C, D are the vertices of a complete quadrangle, there is one and only one planar net of rationality containing them ; and a point P belongs to this net if and only if P is one of a sequence of points ABCDD^D^ , finite in number, such that D l is the intersection of two sides of the original quad- rangle and such that every other point D t is the intersection of two lines join- ing pairs of points of the set ABCDD^ A-i- 34. Nets of rationality in space. DEFINITION. A point is said to be rationally related to two planar nets R 2 , R| in different planes but having a linear net in common, provided it is the intersection of two lines each, of which joins a point of R 2 to a distinct point of R 2 2 . A line is said to be rationally related to R 2 , R|, if it joins two, a plane if it joins three, points which are rationally related to them. The set of all points, lines, and planes rationally related to R 2 , R| is called the net of rationality in space (or of three dimensions) determined by RI > R% 5 ft is a l so called the spatial net defined by R 2 , R 2 2 . Theorems analogous to those derived for planar nets may now be derived for nets of rationality in space. We note first that every point of R 2 and of R * is a point of the spatial net R 3 defined by R x 2 , R * (the definition applies equally well to the points of the linear net common to R 2 , R|) ; and that no other points of the planes of these planar nets are points of R 3 . The proofs of the fundamental theorems of align- ment, etc., for spatial nets can, for the most part, be readily reduced to theorems concerning planar nets. We note first : LEMMA. Any line joining a point A^ of R? to a distinct point P of R 3 meets R 2 . (A, E) Proof. By hypothesis, through P pass two lines, each of which joins a point of R 2 to a distinct point of R 2 2 . We may assume these lines distinct from the line PA V since otherwise the lemma is proved. Let the two lines through P meet R a 2 , R| in B v B 2 and C v C z respec- tively (fig. 40). If A v B v C l are not collinear, the planes PA^ and PA^ meet R 2 in the lines A^B^ and A^ respectively, which meet the linear net common to R 2 , R 2 in two points S, T respectively (Theorems 11, 12). The same planes meet the plane of R 2 2 in the lines SB 2 and TC^ respectively, which are lines of R 2 , since S 9 T are points of R 2 2 . These lines meet in a point A 2 of R 2 2 (Theorem 11), which is evidently the point in which the line PA meets the plane of R 2 2 . If A v B v C^ are collinear, let J 2 be the intersection of PA^ with the 90 THE FUNDAMENTAL THEOEEM [CHAP. IV plane of R 2 2 , and S the intersection of A l B l with the linear net common to R? and R 2 2 . Since A 1 is in R (SBfiJ, the perspectivity SC 1 B l A = SC^Ao implies that A 2 is in R(SB 2 C 2 ) and hence in R 3 2 . m P Fio. 40 THEOREM 13. Any line of the spatial net R 3 defined by R*, R| meets R*andR*. (A, E) FIG. 41 Proof. By definition the given line I contains two points A and B of the net R 8 (fig. 41). If A or B is on Rf or R|, the theorem reduces to the lemma. If not, let ^ be a point of R.?, and A z and B 2 the points in which, by the lemma, %A and P^B meet R 2 2 ; also let JJ' be any 84] STETS OF RATIONALITY 91 point of R x 2 not in the plane P^AB, and let P^A and P^B meet R 3 2 in A and .#2". The lines J 2 2 and -.4^| meet in a point of R 2 2 (Theorem 11), and this point is the point of intersection of I with the plane of R|. The argument is now reduced to the case considered in the lemma. THEOREM 14 The points of a spatial net lying on a line of the spatial net form a linear net. (A, E) Proof. Let I be the given line, R 2 and R 2 2 the planar nets defining the spatial net R 3 , and L : and L 2 the points in which (Theorem 13) I meets Rf and R 2 2 (L t and L 2 may coincide). Let A^ be any point of R 2 not on I or on R 2 , and S the point in which A^L^ meets the linear net common to R 2 and R 2 2 (fig. 42). If L^ and L 2 are distinct, the lines PIG. 42 FIG. 43 SL t and SL 2 meet R 2 and R 2 2 in linear nets (Theorem 12); and, by Theorem 13, a line joining any point P of R 3 on I to A l meets each of these linear nets. Hence all points of R 3 on I are in the planar net determined by these two linear nets. Moreover, by the definition of R 8 , all the points of the projection from A of the linear net on SL 2 upon I are points of R 3 . Hence the points of R 3 on I are a linear net. If x = ,= S, then, by definition, there is on I a point A of R*, and the line AA^ meets R| in a point A z (fig. 43). The lines SA^ and SA 2 meet R? and R| in linear nets R x and R 2 by Theorem 12. If JB X is any point of R x other than Jt p the line .A^ meets R 2 2 in a point J? 2 by Theorem 13. By Theorem 12 all points of I in the planar net deter- mined by Rj and R 2 form a linear net, and they obviously belong to R 3 . Moreover, any point of R 3 on I, when joined to A v meets R 2 2 by Theo- rem 13, and hence belongs to the planar net determined by R x and R 2 . Hence, in this case also, the points of R 8 on I constitute a linear net. 92 THE FUNDAMENTAL THEOEEM [CHAP, iv THEOREM 15. The points and lines of a spatial net R 3 which lie on a plane a of the net form a planar net. (A, E) Proof. By definition a contains three noncollinear points A, B, C of R 8 , and the three lines AB, BC, CA meet the planar nets R* and R 2 2 , which determine R 3 , in points of two linear nets R x and R 2 , consisting entirely of points of R 3 . These linear nets, if distinct, determine a planar net R 2 in a, which, by Theorem 10, consists entirely of points and lines of R 3 . Moreover, any line joining a point of R 3 in a to A or B or C must, by Theorem 13, meet R x and R 2 and hence be in R\ Hence all points and lines of R 3 on a are points and lines of R 2 . This completes the proof except in case R t = R a , which case is left as an exercise. COROLLARY 1. A net of rationality in space is a space satisfying Assumptions A and E, if "line" le interpreted as "linear net" and "plane" as "planar net." (A, E) For all assumptions A and E, except A 3, are evidently satisfied ; and A 3 is satisfied because there is a planar net of points through any three points of a spatial net R 3 , and any two linear nets of this planar net have a point in common. This corollary establishes at once all the theorems of alignment in a net of rationality in space, which are proved in Chap. I, as also the principle of duality. "We conclude then, for example, that two planes of a spatial net meet in a line of the net, and that three planes of a spatial net meet in a point of the net (if they do not meet in a line), etc. Moreover, we have at once the following corollary : COROLLARY 2. A spatial net is determined ly any two of its planar nets. (A, E) EXERCISES 1. If A,B, C,Z>, E are the vertices of a complete space five-point, there is one and only one net of rationality containing them all. A point P belongs to this net if and only if P is one of a sequence of points ABCDEI^I^ , finite 'in number, such that I I is the point of intersection of three faces of the original five-point and every other point J t - is the intersection of three distinct planes through triples of points of the set ABCDEI^ - - - 1^^. 2. Show that a planar net is determined if three noncollinear points and a line not passing through any of these points is given. 3. Under what condition is a planar net determined by a linear net and two points not in this net ? Show that two distinct planar nets in the same plane can have at most a linear net and one other point in common. 34, 35] . THE FUNDAMENTAL THEOREM: 93 4. Show that a set of points in a plane -which is protective with L, planar net is a planar net. 5. A line joining a point P of a planar net to any point not in the net, but on a line of the net not containing P : has no other point than P in common with the net. 6. Two points and two lines in the same plane do not in general belong to the same planar net. 7. Discuss the determination of spatial nets by points and planes, similarly to Exs. 2, 3, and 6. 8. Any class of points projective with a spatial net is itself a spatial net. 9. If a perspective collineation (homology or elation) in a plane with center A and axis I leaves a net of rationality in the plane invariant, the net contains A and /. 10. Prove the corresponding proposition for a net of rationality in space invariant tinder a perspective transformation. 11. Show that two linear nets on skew lines always belong to some spatial net ; in fact, that the number of spatial nets containing two given linear nets on skew lines is the same as the number of linear nets through two given points. 12. Three mutually skew lines and three distinct points on one of them determine one and only one spatial net in which they lie. 13. Give further examples of the determination of spatial nets by lines, 35. The fundamental theorem of projectivity. It has been shown (Chap. Ill) that any three distinct elements of a one-dimensional form may be made to correspond to any three distinct points of a line by a projective transformation. Likewise any four elements of a two-dimensional form, no three of which belong to the same one- dimensional form, may be made to correspond to the vertices of a complete planar quadrangle by a projective transformation ; and any five elements of a three-dimensional form, no four of which belong to the same two-dimensional form, may be made "to correspond to the five vertices of a complete spatial five-point by a projeetive transformation. These transformations are of the utmost importance. Indeed, it is the principal object of projective geometry to discover those prop- erties of figures which remain invariant when the figures are sub- jected to projective transformations. The question now naturally arises, Is it possible to transform any four elements of a one- dimensional form into any four elements of another one-dimensional form ? This question must be answered in the negative, since a har- monic set must always correspond to a harmonic set. The question 94 THE FUNDAMENTAL THEOEEM [CHAP.IV tlien arises whether or not a projective correspondence between one- dimensional forms is completely determined when three pairs of homologous elements are given. A partial answer to this funda- mental question is given in the next theorem. LEMMA 1. If a projectivity leaves three distinct points of a line fixed, it leaves fixed every point of the linear net defined ~by these points. This follows at once from the fact that if three points are left invariant by a projectivity, the harmonic conjugate of any one of these points with respect to the other two must also be left inva- riant by the projectivity (Theorems 2 and 3, Cor.). The projectivity in question must therefore leave invariant every point harmonically related to the three given points. THEOREM 16. THE FUNDAMENTAL THEOREM OF PROJECTIVITY FOR A NET OF RATIONALITY ON A LINE. If A, B, C, D are distinct points of a linear net of rationality, and A r , J3 f , C f are any three distinct points of another or the same linear net, then for any proj'ectivities giving ABCD-^AWC'D* and ABCD^A r r O r D[ 9 we have D J =D[. (A, E) Proof. If TT, TT X are respectively the two pro jectivities of the theorem, the projectivity Tr^" 1 leaves A'JB f C f fixed and transforms D f into D[. Since D f is harmonically related to A f , B r , C r (Theorem 7), the theorem follows from the lemma. This theorem gives the answer to the question proposed in its relation to the transformation of the points of a linear net. The corresponding proposition for all the points of a line, ie. the prop- osition obtained from the last theorem by replacing " linear net " by "line," cannot be proved without the use of one or more additional assumptions (cf. 50, Chap. VI). We have seen that it is equiva- lent to the proposition: If a projectivity leaves three points of a line invariant, it leaves every point of the line invariant. Later, by means of a discussion of order and continuity (terms as yet unde- fined), we shall prove this proposition. This discussion of order and continuity is, however, somewhat tedious and more difficult than the rest of our subject ; and, besides, the theorem in question is true in spaces,* where order and continuity do not exist. It has * Different, of course, from ordinary space; " rational spaces*' (cf. p. 98 and the next footnote) are examples in which continuity does not exist; u finite spaces," of which examples are given in the introduction ( 2), are spaces in which neither order nor continuity exists. 35] THE FUNDAMENTAL THEOREM 95 therefore seemed desirable to give some of the results of this theorem before giving its proof in terms of order and continuity. To this end we introduce here the following provisional assumption of projectivity, which will later be proved a consequence of the order and continuity assumptions which will replace it. This provisional assumption may take any one of several forms. "We choose the fol- lowing as leading most directly to the desired theorem : AN ASSUMPTION OF PROJECTIVITY : P. If a projectivity leaves each of three distinct points of a line- invariant, it leaves every point of the line invariant* We should note first that the plane and space duals of this assump- tion are immediate consequences of the assumption. The principle of duality, therefore, is still valid after our set of assumptions has been enlarged by the addition of Assumption P. "We now have : THEOEEM 17. THE FUNDAMENTAL THEOREM OF PROJECTTTE GEOM- ETRY, f If 1, @, 3,4- are any four elements of a one-dimensional primitive form, and l f , % ? , 3 l are any three elements of another or the same one- dimensional primitive form, then for any projectivities giving 134~^ 1W&4' and 1^34^1 r ^3 f 4[ } we have 4'=4[. (A, E, P) Proof. The proof is the same under the principle of duality as that of Theorem 16, Assumption P replacing the previous lemma. This theorem may also be stated as follows : A projectivity 'between one-dimensional primitive, forms is uniquely determined when three pairs of homologous elements are given. (A, E, P) COROLLARY. If tico pencils of points on different lines are projectile and have a self-corresponding point, they are perspective. (A, E, P) * We have seen in the lemma of the preceding theorem that the projectivity described in this assumption leaves invariant every point of the net of rationality defined by the three given points. The assumption simply states that if all the points of a linear net remain invariant under a protective transformation, then all the points of the line containing this net must also remain invariant. It will be shown later that in the ordinary geometry the points of a linear net of rationality on a line corre- spond to the points of the line whose coordinates, when represented analytically, are rational numbers. This consideration should make the last assumption almost, if not quite, as intuitionally acceptable as the previous Assumptions A and E. t On this theorem and related questions there is an extensive literature to which references can be found in the Encyklopadie articles on Protective Geometry and Foundations of Geometry. It is associated with the names of von Staudt, Klein, Zeuthen, Liiroth, Darboux, F. Schur, Pieri, Wiener, Hilbert. Cf . also 50, Chap. VI. 96 THE FUNDAMENTAL THEOREM [CHAP.IV Proof. For if is the self-corresponding point, and AA f and BB' are any two pairs of homologous points distinct from 0, the perspec- tivity whose center is the intersection of the lines A A', BB 1 is a projectivity between the two lines which has the three pairs of homologous points 00, AA f , BB f , which must be the projectivity of the corollary by virtue of the last theorem. The corresponding theorems for two- and' three-dimensional forms are now readily derived. We note first, as a lemma, the propositions in a plane and in space corresponding to Assumption P. LEMMA 2. A projectile transformation which leaves invariant each * j. * four > j. ,c a plane three , . , , 7 , ^ of a set of - points of * no of WIMMI oelonq to me same J J five * J space four J y line ^ . . . . _, - the plane. , . , leaves invariant every point of (A. K P) plane J * J space. ^ ' Proof. If A, B, C, D are four points of a plane no three of which are collinear, a projective transformation leaving each of them in va- riant 'must also leave the intersection of the lines AB, CD invariant. By Assumption P it then leaves every point of each of the lines AB, CD invariant. Any line of the plane which meets the lines AB and CD in two distinct points is therefore invariant, as well as the inter- section of any two such lines. But any point of the plane may be determined as the intersection of two such lines. The proof for the case of a projective transformation leaving invariant five points no four of which are in the same plane is entirely similar. The existence of perspective collineations shows that the condition that no three (four) of the points shall be on the same line (plane) is essential. THEOREM 18. A projective collineation* "between two planes (or within a single plane) is uniquely determined when four pairs of homologous points are given, provided no three of either set of four points are collinear. (A, E, P) Proof. Suppose there were two collineations TT, TT X having the given pairs of homologous points. The collineation irjr~ l is then, by the lemma, the identical collineation in one of the planes. This gives at once TTj = TT, contrary to the hypothesis. * We confine the statement to the case of the collineation for the sake of sim- plicity of enunciation. Projective transformations which are not collineations will be discussed in detail later, at which time attention will be called explicitly to the fundamental theorem. 35] THE FUSTDAMEXTAL THEOBEM 97 By precisely similar reasoning we have : THEOREM 19. A protective collineation in space is uniquely deter- mined when five pairs of liomoloyous points are given, provided no four of either set of five points are iii the same plane. (A, E, P) The fundamental theorem deserves its name not only because so large a part of projective geometry is logically connected with it, but also because it is used explicitly in so many arguments. It is indeed possible to announce a genial course of procedure that appears in the solution of most " linear " problems, i.e. problems which depend on constructions involving points, lines, and planes only. If it is desired to prove that certain three lines l v 1 2 , / 3 pass through a point, find two other lines m v m^ such that the four points mj v mj^ mJ ZJ m^n^ may be shown to be projective with the four points mj v mj^, mj,^ mjn^ respectively. Then, since in this projectivity the point m^n^ is self- corresponding, the three lines l v l, 1 B joining corresponding points are concurrent (Theorem 17, Cor.). The dual of this method appears when three points are to be shown collinear. This method may be called the principle of projectivity, and takes its place beside the principle of duality as one of the most powerful instruments of pro- jective geometry. The theorems of the next section may be regarded as illustrations of this principle. They are all propositions from which the principle of projectivity could be derived, ie. they are propositions which might be chosen to replace Assumption P. We have already said that ordinary real (or complex) space is a space in which Assumption P is valid. Any such space we call a properly projective space. It will appear in Chap. VI that there exist spaces in which this assumption is not valid. Such a space, ie. a space satisfying Assumptions A and E but not P, we will call an improperly projective space. From Theorem 15, Cor. 1 and Lemma 1, we then have THEOREM 20. A net of rationality in space is a properly projective space. (A, E) It should here be noted that if we added to our list of Assump- tions A and E another assumption of closure, to the effect that all points of space belong to the same net of rationality, we should obtain a space in which all our previous theorems are valid, in- cluding the fundamental theorem (without using Assumption P). 98 THE FUNDAMENTAL THEOREM [CHAP. IV Such a space may be called a rational space. In general, it is clear that any complete five-point in any properly or improperly protective space determines a subspace which is rational and therefore properly projective. 36. The configuration of Pappus. Mutually inscribed and circum- scribed triangles. THEOREM 21. If A, B, C are any three distinct points of a line I, and A\ B\ C f any three distinct points of another line I' meeting I, the three points of intersection of the pairs of lines AB r and A!B> BC 1 and B'C, CA 1 and C f A are collinear. (A, E, P) Proof. Let the three points of intersection referred to in the theorem be denoted by C r ", A", B" respectively (fig. 44). Let the line "C" meet the line B'C in a point D (to be proved identical with A"); also let B"C" meet V in A v the line A'B meet AC r in B v the line AB' meet A r C in B[. We then have the following perspectivities : 4 By the principle of projectivity then, since in the projectivity thus established C n is self-corresponding, we conclude that the three lines A^A'i B n B v DB meet in the point '. Hence D is identical with A n , and A", B ff , C" are collinear. It should be noted that the figure of the last theorem is a con- figuration of the symbol COXFIGUEATICXtf OF PAPPUS 99 It is known as the concur crtiajL. of Pappus* It should also be noted that this configuration may be considered as a simple plane hexagon (six-point) inscribed in two intersecting lines. If the sides of such a hexagon be denoted in order by 1, 2, 3, 4, 5, 6, and if we call the sides 1 and 4 opposite, likewise the sides 2 and 5, and the sides 3 and 6 (cf. Chap. II, 14), the last theorem may be stated in the following form : COROLLARY. If a simple hexagon le inscribed in two intersecting lines, the three pairs of opposite sides will intersect in collinear points.^ Finally, we may note that the nine points of the configuration of Pappus may be arranged in sets of three, the sets forming three triangles, 1, 2, 3, such that 2 is inscribed in 1, 3 in 2, and 1 in 3. This observation leads to another theorem con- nected with the Pappus configuration. THEOREM 22. If A^B^C^ be a triangle inscribed in a triangle p iG 45 A^B! V there exists a certain set of triangles each of which is inscribed in the former and circumscribed about the latter. (A, E, P) Proof. Let \a\ be the pencil of lines with center A^ [&] the pencil with center B^ and [c] the pencil with center C^ (fig. 45). Consider the perspectivities [a] 2 [&] == [c]. In, the projectivity thus estab- lished between [a] and [c] the line A l C l is self-corresponding; the pencils of lines [a], [c] are therefore perspective (Theorem 17, Cor. (dual)). Moreover, the axis of this perspectivity is C 2 A^ for the lines Aft* and C^ are clearly homologous, as also the lines A^A Z and O^A^. Any three homologous lines of the perspective pencils [a], [&], [c] then form a triangle which is circumscribed about A 1 B 1 C 1 and inscribed in A^B^C^ * Pappus, of Alexandria, lived about 340 A.D. A special case of this theorem may be proved without the use of the fundamental theorem (cf. Ex. 3, p. 52). t In this form it is a special case of Pascal's theorem on conic sections (cf. Theorem 3, Chap. V). 100 THE FUNDAMENTAL THEOREM [CHAP. IV EXERCISES 1. Given a triangle ABC and two distinct points A', Z>'; determine a point C" sucli that the lines A A', BB', CC'are concurrent, and also the lines A I?, BC', CA' are concurrent, i.e. such that the two triangles are perspective from two dif- ferent points. The two triangles are there said to he doubly perspective. 2. If two triangles ABC and A'B'C' are doubly perspective in such a way that the vertices A, B, C are homologous with A', B', C' respectively in one perspectivity and with J3', C', A' respectively in the other, they will also be per- spective from a third point in such a way that A, B, C are homologous respec- tively with C', A', B'^ i.e. they will be triply perspective. 3. Show that if A", B", C" are the centers of perspectivity for the triangles in Ex. 2, the three triangles ABC, A'B'C', A"B"C" are so related that any two are triply perspective, the centers of perspectivity being in each case the vertices of the remaining triangle. The nine vertices of the three triangles form the points of one configuration of Pappus, and the nine sides form the lines of another configuration of Pappus. 37. Construction of projectivities on one-dimensional forms. THEOREM 23. A necessary and sufficient condition for the projectivity on a line MNAB -^ MNA'B f ( 31 * JV) is Q(3IAB, NB'A'). (A, E, P) Proof. Let n be any line on* N not passing through A (fig. 46). Let O l be any point not on n or on MA, and let A l and B t be the intersections respectively of O t A and O^B with n. Let 2 be the intersection and B'B V Then 0, 0, By Theorem 17 the projectivity so determined on the line AM is the same as The only possible double points of the projectivity are N and the intersection of AN with O a 2 . Hence 0^ passes through M 9 and Q(MAB, NB f A f ) is determined by the quadrangle O^A^ 37] ONE-DIMENSIONAL PEOJECTIYITIES 101 Conversely, if Q(1LLB, XJB'A') we have a quadrangle O and hence - and by this construction 31 is self-corresponding, so that If iu the above construction we have J/"= JV, we obtain a projec- tivity with the single double point J/= JV. DEFINITION. A projectivity on a one-dimensional primitive form with a single double element is called parabolic. If the double ele- ment is J/, and AA f 9 BB* are any two homologous pairs, the pro- jectivity is completely determined and is conveniently represented by MMAB ^ MMA'B'. COROLLARY. A necessary and sufficient condition for a parabolic projectivity MHAB-^MMA'B* is Q(3IA, MB'A f ). (A, E, P) THEOREM 24 If we hare Q(ABC, A'3 f C f ), we have also Q(A f B'C r , ABC). Proof. By the theorem above, Q(A3C, A'B'C 1 ) implies AA r BC^ AA'C'B', which is the inverse of A r AB r C f -^ A A OB 9 which, by the theorem above, implies Q(A'B f C' 9 ABC). The notation Q (ABC, A f B f C l ) implies that A, B, C are the traces of a point triple of sides of the quadrangle determining the quadrangular set. The theorem just proved states the existence of another quadrangle for which A\ B r , C 1 are a point triple, and consequently A, B, C are a triangle triple. This theorem therefore establishes the existence of oppositely placed quadrangles, as stated in 19, p. 50. This result can also be propounded as follows: THEOREM 25. If two quadrangles I^P 2 P Z P^ and Q^^Q^Q^ are so related J? to Q v J% to Q y etc. that five of the sides l$I$(i,j =1,2,8,4; i =/) meet the five sides of the second which are opposite to Q t Qj in points of a line I, the remaining sides of the two quadrangles meet on L (A,E,P) 102 THE FUNDAMENTAL THEOEE3M [CHAP. IV Proof. The sides of the first quadrangle meet I in a quadrangular set Q (P 12 P 13 P^ P*Pu&) ; hence Q (P 2 PA P*PM* But, by hypoth- esis, five of the sides of the second quadrangle pass through these points as follows : Q^ through J^, Q^Q 9 through P^ 9 Q& through P^> Q z Qt through 2J a , QQ 2 through P is , Q 2 Q Z through P u . As five of these conditions are satisfied, by Theorem 3, Chap. II, they must all be satisfied. EXERCISES 1. Given one double point of a projectivity on a line and two pairs of homologous points, construct the other double point. 2. If a, b, c are three nonconcurrent lines and A', B', C' are three collinear points, give a construction for a triangle whose vertices A. , B, C are respectively on the given lines and whose sides BC, CJL, AB pass respectively through the given points. What happens when the three lines a , I, c are concurrent ? Dualize. 38. Involutions, DEFINITION. If a projectivity in a one-dimensional form is of period two, it is called an involution. Any pair of homol- ogous points of an involution is called a conjugate pair of the involution or a pair of conjugates. It is clear that if an involution transforms a point A into a point A r , then it also transforms A r into A] this is expressed by the phrase that the points A, A 1 correspond to each other doully. The effect of an invo- lution is then simply a pairing of the elements of a one-dimensional form such that each element of a pair corresponds to the other ele- ment of the pair. This justifies the expression "a conjugate pair" applied to an involution. THEOREM 26. If for a single, point A of a line which is not a double point of a projectivity TT on the. line we have the relations 7r(A)t=A ! and 7r(A') = A, the projectivity is an involution. (A, E, P) Proof. For suppose P is any other point on the line (not a double point of ?r), and suppose 7r(P) = P f . There then exists a projectivity giving AA'PP 1 ^ A'AP'P (Theorem 2, Chap. III). By Theorem 17 this projectivity is TT, since it has the three pairs of homologous points A, A*\ A r , A\ P, P r . But in this projectivity P r is transformed into P. Thus every pair of homologous points corresponds doubly. COROLLARY. An involution is completely determined when two pairs of conjugate points are given. (A, E, P) INVOLUTIONS 103 THEOREM 27. A necessary and sufficient condition that three pairs of points A, A r ; B, B' ; C, C 1 be conjugate pairs of an involution is Q(ABC,A f B'C f ). (A, E, P) Proof. By hypothesis we have AA'BC-^A'AB'C'. By Theorem 2, Chap. Ill, we also have A'AS'C'-fiA which, with the first projectivity, gives A necessary and sufficient condition that the latter projectivity hold is Q(ABC, AWC r ) (Theorem 23). COROLLARY 1. If an involution has doulle points, they are harmonic conjugates with respect to every pair of the involution. (A, E, P) For the hypothesis A~A f ,B B J gives at once H(AB 9 CC f ) as the condition of the theorem. COROLLARY 2. An involution is completely determined when two double points are given, or when one double point and one pair of conjugates are given. (A, E, P) COROLLARY 3. If J/, N are distinct doulle points of a projectivity on a line, and A,A r ; B,B' are any two pairs of homologous elements, the pairs M, N; A, B J ; A 1 , B are conjugate pairs of an involution.* (A, E, P) COROLLARY 4 If an involution has one double element, it has another distinct from the first. (A, E, P) COROLLARY 5. The projectivity ABGD-j^ABDC "between four dis- tinct points of a line implies the relation H (AB, CD). (A, E, P) Eor the projectivity is an involution (Theorem 26) of which A, B are double points. The result then follows from Cor. 1. 39. Axis and center of tiomology. THEOREM 28. If \A] and [B] THEOREM 28'. If [I] and \m\ are any two projectile pencils are any two projectile pencils of of points in the same plane on lines in the same plane on distinct * This relation is sometimes expressed by saying, s l The pairs of points are in involution." From what precedes it is clear that any two pairs of elements of a one-dimensional form are in involution, but in general three pairs are not. 104 THE FUNDAMENTAL THEOEEM [CHAP. IV distinct lines l v 1 2> there exists a line I siich that if A v B I and A^ B z are any two pairs of homologous points of the two pencils, the lines A^ and A^ intersect on I (A,E,P) DEFINITION. The line I is called the axis of homology of the two pencils of points. points S v S Q9 there exists a point S such that if a v \ and a 2> & 2 are any two pairs of homologous lines of the two pencils, the points a^ and a 2 6 1 are collinear with S. (A, E, P) DEFINITION. The point S is called the center of homology of the pencils of lines. Proof. The two theorems being plane duals of each other, we may confine ourselves to the proof of the theorem on the left. From the projectivity [S] ^ [A] follows A^B^B^A] (fig. 47). But in this pro- jectivity the line A^ is self-corresponding, so that (Theorem 17, Cor.) A A & 47 the two pencils are perspective. Hence pairs of corresponding lines meet on a line I ; e.g. the lines A^ s and B^ z meet on I as well as A^B Z and B^Ay To prove our theorem it remains only to show that B 2 A S and A 2 B^ also meet on L But the latter follows at once from Theorem 21, since the figure before us is the configuration of Pappus. COROLLARY. If [A], [B] are not COROLLARY. If [I], [m] are not perspective) the axis of homology is perspective, the center of homology the line joining the points homol- ogous with the point l^ regarded first as a point of l^ and then as a point of l y is the point of intersection of the lines homologous with the line S^ regarded first as a line of [I] and then as a line of [m]. For in the perspectivity J^B] = B l [A'] the line ^ corresponds to ) > and hence the point l^ corresponds to K t in the projectivity [#] 7^ [A], Similarly, ll z corresponds to l^ y 39] OEKTEE AXD AXIS OF HOROLOGY 105 EXERCISES 1. There is one and only one projectivity of a one-dimensional form leaving invariant one and only one element 0, and transforming a given other element -4 to an element B. 2. Two protective ranges on skew lines are always perspective. 3. Prove Cor. 5, Theorem 27, without using the notion of involution. 4. If MNAB-^MyA'B'* then MXAA'- J/JV.RB'. 5. If P is any point of the axis of homology of two projective ranges M A"^' tliel1 tlie P r J ectiv % ^[-4] 77 **[#] is an involution. Dualize. 6. Call the faces of one tetrahedron a ls 03, a 3 , a 4 and the opposite vertices *i 1? -dU, -4 3 , J 4 respectively, and similarly the faces and vertices of another tetra- hedron ft, ft, ft, ft and B 19 B s , B+ If 4i, J a , .4,, ^ lie on ft, ft, ft, ft respectively, and B^ lies on a x , B on a 2 , -5 3 on a 3 < then ^ lies on a 4 . Thus each of the two tetrahedra related in this fashion is both inscribed and cir- cumscribed to the other. 7. Prove the theorem of Desargues (Chap. II) by the principle of pro- jectivity. 8. Given a triangle ABC and a point A\ show how to construct two points B', C' such that the triangles ABC and A'B'C' are perspective from four different centers. 9. If two triangles A^B^ and A^B 2 C Z are perspective, the three points (4 A, A^) = C,, (AiC v A&) = B 9 , (B X C 3 , J5 2 CO = A v if not collinear, form a triangle perspective with the first two, and the three centers of perspectivity are collinear. * 10. (a) If it is a projectivity in a pencil of points [J.] on a line a with inva- riant points A v AV and if [], \M ] are the pencils of points on two lines /, w through A^AQ respectively, show by the methods of Chap. Ill that there exist three points S 19 S 3 , S s such that we have where v (A ) = A' ; that S v S z , -4 2 are collinear ; and that ^ 2 , g , A ^ are collinear. (&) Using the fundamental theorem, show that there exists on the line ^A 2 a point S such that we have (c) Show that (5) could be used as an assumption of projectivity instead of Assumption P ; i.e. P could be replaced by : If TT is a projectivity with fixed points ^t 1? A%, giving 7r(-4) = A' in a pencil of points 4], and [] is a pencil of points on a line I through A 19 there exist two points S v S 2 such that S l S, 106 THE FUNDAMENTAL THEOREM [CHAP. IV * 11. Show that Assumption P could be replaced by the corollary of Theorem 17. * 12. Show that Assumption P could be replaced by the following : If we have a projectivity in a pencil of points defined by the perspecthities [-X] = [] = [A"], and [JIT] is the pencil of points on the line 5^, there exist on the base of [] two points Si, S such that we have also =[*'] 40. Types of collineations in the plane. We have seen in the proof of Theorem 10, Chap. Ill, that if O^O, is any triangle, there exists a collineation II leaving O v 2 , and O s invariant, and trans- forming any point not on a side of the triangle into any other such III FIG. 48 point. By Theorem 18 there is only one such collineation II. By the same theorem it is clear that II is fully determined by the projec- tivity it determines on two of the sides of the invariant triangle, say 2 3 and 1 S . Hence, if H x is a homology with center O l and axis 2 O st which determines the same projectivity as II on the line O^, and if H 2 is a homology with center 3 and axis 1 8 , which deter- mines the same projectivity as II on the line 8 , then it is evident that <! TYPES OF COLLIXEATIOXS 107 * It is also evideut that no point not a vertex of the invariant triangle can be fixed unless II reduces to a homology or to the identity. Such a transformation II when it is not a hornology is said to be of Type I 9 and is denoted by Diagram / (fig. 48). EXERCISE Prove that two honiologies with, the same center and axis are commutative, and hence that two projectivities of Type I with the same invariant figure are commutative. Consider the figure of two points 1? 2 and two lines o v 2 , such that 1 and 2 are on o v and o l and 2 are on O r A collineation II which is the product of a homology H, leaving 3 and o. 2 invariant, and an elation E, leaving O l and o l invariant, evidently leaves this figure invariant and also leaves invariant no other point or line- If A and B are two points not on the lines of the invariant figure, and we require that TL(4\=sB this fixes the transformation (with two distinct double lines) among the lines at O v and the parabolic transformation among the lines at 2 , and thus determines II completely. Clearly if II is not to reduce to a homology or an elation, the line AB must not pass through O l or 2 . Such a transformation II, when it does not reduce to a homology or an elation or the identity, is said to be of Type II and is denoted by Diagram II (fig. 48). EXERCISE Two projective collineations of Type IT, having the same invariant figure, are commutative. DEFINITION The figure of a point and a line o on is called a lineal element Oo. A collineation having a lineal element as invariant figure must effect a parabolic transformation both on the points of the line and on the lines through the point. Suppose Aa and Sb are any two lineal ele- ments whose points are not on o or collinear with 0, and whose lines are not on or concurrent with o. Let E x be an elation with center and axis OA 9 which transforms the point (oa) to the point (ol). Let E 2 be an elation of center (AB, o) and axis o, which transforms A to B. Then II = E 2 E X has evidently no other invariant elements than and o and transforms Aa to BL 108 THE FUNDAMENTAL THEOEEM [CHAP. IV Suppose that aaother projeetivity II' would transfer Aa to Bl with Oo as only invariant elements. The transformation II 7 would evidently have the same effect on the lines of and points of o as II. Hence II'II- 1 would he the identity or an elation. But as ITrT 1 ^) = . it would be the identity. Hence II is the only projectivity which trans- forms Aa to b with Oo as only invariant. A transformation having as invariant figure a lineal element and no other invariant point or line is said to be of Type III, and is denoted by Diagram /// (fig. 48). A homology is said to be of Type /Fand is denoted by Diagram IV. An elation is said to be of Type V and is denoted by Diagram F. It will be shown later that any collineation can be regarded as be- longing to one of these five types. The results so far obtained may be summarized as follows : THEOREM 29. A protective collineation with given invariant figure F, if of Type I or II will transform any point P not on a line of F into any other such point not on a line joining P to a point of F; if of Type III will transform any lineal element Pp such that p is not on a point, or P on a line, of F into any other such element Qq; if of Type IV or V, will transform any point P into any other point on the line joining P to the center of the collineation. The r61e of Assumption P is well illustrated by this theorem. In case of each of the first three types the existence of the required collineation was proved by means of Assumptions A and E, together with the existence of a sufficient number of points to effect the construction. But its uniqueness was established only by means of Assumption P. In case of Types IV and F, both existence and uniqueness follow from Assumptions A and E. EXERCISES 1. State the dual of Theorem 29. 2. If the number of points on a line is^? + 1, the number of collineations with a given invariant figure is as follows : Type/, O-2)O~3). Type//, o- 2) (>-_!), Type m,p(p~ I)*. Type/F, p~2. TypeF, p-1. In accordance with the results of this exercise, when the number of points on a line is infinite it is said that there are oo 2 transformations of Type / or //; oo 3 of Type ///; and oo 1 of Types /Fand F. CHAPTER V* CONIC SECTIONS 41. Definitions. Pascal's and Brianchon's theorems. DEFINITION. The set of all points of intersection of homologous lines of two protective, nonperspective flat pencils which are on the same plane but not on the same point is called a point conic (fig. 49). The plane dual of a point conic is called a line conic (fig. 50). The space dual of a point conic is called a cone of planes; the space dual . 49 FIG. 50 of a line conic is called a cone of lines. The point through which pass all the lines (or planes) of a cone of lines (or planes) is called the vertex of the cone. The point conic, line conic, cone of planes, and cone of lines are called one-dimensional forms of the second degree.^ The following theorem is an immediate consequence of this defi- nition. THEOREM 1. The section of a cone of lines ly a plane not 6n the vertex of the cone is a point conic. The section of a cone of planes by a plane not on the vertex is a line conic. Now let A^ and B l be the centers of two flat pencils defining a point conic. They are themselves, evidently, points of the conic, for the line A^BI regarded as a line of the pencil on A 1 corresponds to some other line through B^ (since the pencils are, by hypothesis, protective * All the developments of this chapter are on the basis of Assumptions A, E, F, and H . t A fifth one-dimensional form a self-dual form of lines in space called the regulus will be defined in Chap. XL This definition of the first four one-dimen- sional forms of the second degree is due to Jacob Steiner (1796-1863). Attention will be called to other methods of definition in the sequel. 109 110 CONIC SECTIONS [CHAP. V but not perspective), and the intersection of these homologous lines is B r The conic is clearly determined by any other three of its points, say A 2 , B z , <7 2 , because the projectivity of the pencils is then determined by (AJ*&) 7: B l (AJ}&) (Theorem 17, Chap. IV). Let us now see how to determine a sixth point of the conic on a line through one of the given points, say on a line Z through B y If the line I is met by the lines A^, A^C^ B^A^ B^G Z in the points S> T, U, A FIG. 51 -j^ UB^A. The other respectively (fig. 5 1), we have, by hypothesis, S double point of this projectivity, which we will call C v is given by the quadrangular set Q^ST, C^AU) (Theorem 23, Chap. IV). A quad- rangle which determines it may be obtained as follows : Let the lines A^B^ and A^BZ meet in a point (7, and the lines AC and A^C^ in a point B ; then the required quadrangle is A^A^CB^ and C l is determined as the intersection of AJB with L C l Trill coincide with 2 , if an d Q ^J *f B is on A 2 B^ (fig. 52). This means that A C, A^Cty and A^B 2 are concurrent in B. In other words, A must be the point of intersection of B X C 2 with the line joining C = (A^B^ (A^) and B = (JljCy^o&j)* an< i I must be the line joining B z anil A . This gives, then, a construction for a line which meets a given conic in only one point. The result of the preceding discussion may be summarized as follows : TJie four points A z , B 2 , <7 2 , C l are points of a point conic HI] PASCAL'S THEOREM 111 determined ly two projectile pencils on A^ and B v if and only if the three points C' = (.4^,) (A&), B = (A^C.) (A 2 C^, A = (B^) (B.,C\) are collinear. The three points in question are clearly the intersections of pairs of opposite sides of the simple hexagon A^B^A^B^C y Since A v B v C^ may be interchanged with -.4.,, J5 2 , C 2 respectively in the above statement, it follows that A v Z? 1? C v C n are points of a conic determined by protective pencils on A 2 and B 2 . Thus, if C^ is any point of the first conic, it is also a point of the second conic, and vice versa. Hence we have established the following theorem : THEOREM 2. STEIXER'S THEOREM. If A and B are any two given points of a conic, and P is a variable point of this conic, ice have In view of this theorem the six points in the discussion may be regarded as any six points of a conic, and hence we have THEOREM 3. PASCAL'S THEOREM.* The necessary and sufficient con- dition that siQTjwwsno tJiree of u'hicJi are collinear, be points of the same conic is that the three pairs of opposite sides of a simple hexagon of ivhicli they are vertices shall meet in collinear points.^ The plane dual of this theorem is ^ THEOREM 3'. jteiAXCHoy's THEOREM. Tlie necessary and sufficient condition that six lines, no three of ichich are concurrent, be lines of a line conic is that the lines joining the three pairs of opposite vertices of any simple hexagon of whicli the given lines are sides, shall be concurrent.^ As corollaries of these theorems we have COROLLARY 1. A line in the plane of a point conic cannot have more than two points in common with the conic. COROLLARY 1'. A point in the plum of a line conic cannot be on more than two lines of the conic. * Theorem 3 was proved by B. Pascal in 1640 when only sixteen years of age. He proved it first for the circle and then obtained it for any conic by projection and feection. This is one of the earliest applications of this method. Theorem 8' was first given by C. J. Brianchon in 1806 (Journal de 1'ficole Polytechnique, Vol. VI, p. 301). t The line thus determined by the intersections of the pairs of opposite sides of any simple hexagon whose vertices are points of a point conic is called the Pascal line of the hexagon. The dual construction gives rise to the Brianchon poM of a hexagon whose sides belong to a line conic. 112 COOTC SECTIONS [OHAP.V Also as immediate corollaries of these theorems we have THEOREM 4. There is one and only one point conic containing Jim given points of a plane no three of which are collinear. THEOREM -4'. There is one and only one line conic containing Jive given lines of a plane no three of which are concurrent. EXERCISES 1. What are the space duals of the above theorems? 2. Prove Brianehon's theorem without making use of the principle of duality. 3. A necessary and sufficient condition that six points, no three of "which are collinear, be points of a point conic, is that they be the points of inter- section (&'), (fa'), (ca'), (a') ( c&/ )> ( a O f tne si des #j &> c an< i a/ ? &'> c/ f two perspective triangles, in which a and a', "b and &', c and c' are homologous. 42. Tangents. Points of contact. DEFINITION. "A line p in the plane of a point conic which meets the point conic in one and only one point P is called a tangent to the point conic at P. A point P in the plane of a line conic through which passes one and only one line p of the line conic is called a point of contact of the line conic on'jp. THEOREM 5. Through any point of a point conic there is one and only one tangent to the point conic. Proof. If PQ is the given point of the point conic and P is any other point of the point conic, while P is a variable point of this conic, we have, by Theorem 2, Any line through P Q meets its homologous line of the pencil on 7^ in a point distinct from P, except when its homologous line is P^. Since a projectivity is a one-to-one correspondence, there is only one line on -ZJ which has I(P^ as its homologous line. THEOREM 5'. 0% any line of a line conic there is one and only one point of contact of the line conic. This is the plane dual of the preceding theorem. EXERCISE Give the space duals of the preceding definitions and theorems. Returning now to the construction in the preceding section for the points of a point conic containing five given points, we recall that 42] TA2s T GE]N T TS 113 the point of intersection Ci of a line I through B was determined by the quadrangular set Q(J3 Z ST, C^U). The points and O l can, by the preceding theorem, coincide on one and only one of the lines through B z * For this particular line Z, A becomes the intersection . 52 of the tangent at B z with I, and the collinearity of the points A, B 9 C may be stated as follows : THEOREM 6. If the vertices of a simple plane five-point are points of a point conic, the tangent to the point conic at one of the vertices meets the opposite, side in a point collinear with the points of inter- section of the other two pairs of nonadjacent sides. This theorem, by its derivation, is a degenerate case of Pascal's theorem. It may also be regarded as a degenerate case in its state- ment, if the tangent be thought of as taking the place of one side of the simple hexagon. It should be clearly understood that the theorem has been obtained by specializing the figure of Theorem 3, and not by a continuity argument. The latter would be clearly impossible, since our assumptions do not require the conic to contain more than a finite number of points. Theorem 6 may be applied to the construction of a tangent to a point conic at any one of five given points P^ P%, P%, P^ jg of the point conic (fig. 53). By this theorem the tangent p l at JJ must be * As explained in the fine print on page 110, this occurs when I passes through the point of intersection of BiOz with the line joining C (A^Bz) (A%Bi) and 114 CONIC SECTIONS [CHAP. V such that the points 2\(P^) = -i (%%) (%**) - B > and (%P 3 ) (P B P^ = C are collinear. But B and C are determined by ^, P%, ^, -^, J^, and hence p l is the line joining ^ to the intersection of the lines BC and P & %. Cl FIG. 53 ' In like manner, if P v P%, P^, P, and jp 1 are given, to construct the point P$ on any line I through P of a point conic containing P^ P Z) P SJ P and of which p l is the tangent at P 19 we need only determine the points A =1\(P Z P^ B = I (JJJ?), and C = (AB) (P 2 P 3 ) ; then P^C meets f in ^ (fig. 53). FIG. 54 In case I is the tangent p 4 at P^ P 5 coincides with ^ and the fol- lowing points are collinear (fig. 54) : 42] TAXGEXTS 115 Hence we have the following theorem : THEOREM 7. If the vertices P 13 P 2 , %, P of a simple quadrangle are points of a point conic, the tangent at % and the side P S P, the tangent at P and the side JJJEJ, and the pair of sides JJJ* and RE. meet in three collinear points. If %, , P B , P 5 and the tangent p l at P are given, the construction determined by Theorem -3 for a point P of the point conic on a line / through P 3 is as follows (fig. 53): Determine C-=(P 1 P 5 )(P 2 P B ), A = pj, and B = (A C)(P^ ; then P,B meets I in %. In case I is the tangent at P z , P coincides with P 3 and we have the result that C = (P 1 J)(P 2 P B ), A^p^p,, B = (P 1 K)(P 5 P) are collinear points, which gives PIG. 5o THEOREM 8. If the vertices of a complete quadrangle are points of a point conic 9 the tangents at a pair of vertices meet in a poi%t of the line joining the diagonal points of the quadrangle which are not on the side joining the two vertices (fig. 55). The last two theorems lead to the construction for a point conic of which there are given three points and the tangents at two of them. Eeverting to the notation of Theorem 7 (fig. 54), let the given points be P, P P z and the given tangents be p^ p r Let I be any line through P y If PZ is the other point in which I meets the point conic, the points A^p^(P^ 9 B^p^(P^, and C =(%%)(%%) are collinear. Hence, if C^l(P^ and B=p(AC), then F 2 is the intersection of I with B%. In case I is the tangent p z at JJ, the points JEJ and P B coincide, and the points 116 CONIC SECTIONS [CHAP.V are collinear. Hence the two triangles P^P^ and p^sPt are per- spective, and we obtain as a last specialization of Pascal's theorem (fig. 56) THEOREM 9. A trianyle whose vertices are points of a point conic is perspective with the triangle formed "by the tangents at these points, the tangent at any vertex "being ho7nologous with the side of the first triangle which does not contain this vertex. COROLLARY. If I(, J^, P are three points of a point conic, the lines P^PV P$P are harmonic with the tangent at P z and the line joining P% to the intersection of the tangents at jfj and P. Proof. This follows from the definition of a harmonic set of lines, on considering the quadrilateral P^A, AB> BP^ PJ( (fig. 56). FIG. 56 43. The tangents to a point conic form a line conic. If P v P^ P 8 , P are points of a point conic and p v p Z9 p s , p are the tangents to the conic at these points respectively, then (by Theorem 8) the line join- ing the diagonal points (JJ JJ) (P B I%) and (JfJJIJ) (J%I) contains the inter- section of the tangents p v p s and also the intersection of p 2> p^. This line is a diagonal line not only of the quadrangle Jf^J^, but also of the quadrilateral PiPzPsP^ Theorem 8 may therefore be stated in the form: THEOREM 10. Tlu complete quadrangle formed ly four points of a point conic and the complete quadrilateral of the tangents at these points have the same diagonal triangle. Looked at from a slightly different point of view, Theorem 8 gives also THEOREM 11. The tangents to a point conic form a line conic. 43] TAXGEXTS 117 Proof. Let JJ, P 2 , P 3 be any three fixed points on a conic, and let P be a variable point of this conic. Let p v p*> Pv P be respectively the tangents at these points (fig. 57). By the corollary of Theorem 28, Chap. IV, JJJE} is the axis of homology of the projectivity between the pencils of points on p 1 and p 2 defined by But by Theorem 10, if Q=(P 1 P 2 ) (P Z P), the points pj> zi PiP& and Q are collinear. For the same reason the points p z p z , pp v Q are collinear. It follows, by Theorem 28, Chap. IV, that the homolog of the variable PIG. 57 point Pip is p 2 p> i.e. p is the line joining pairs of homologous points on the two lines p v p^ so that the totality of the lines p satisfies the definition of a line conic. COROLLARY. The center of homology of the projectivity P l [P] -^ P% [P] determined "by the points P of a point conic containing P^ 1^ is the intersection of the tangents at 1^ P^. The axis of honwlogy of the projectimty p l [jp] -^ p% [p] determined by the lines p of a line conic containing the lines p lt p 2 is the line joining the points of contact f Pv P* THEOREM 12. If J^ is a fixed and P a variable point of a point conic, and p v p are the tangents at these two points respectively, then 118 CONIC SECTIONS [CHAP.V Proof. Using the notation of the proof of Theorem 11 (fig. 57), we have where Q is always on JJJJ. But we also have and, by Theorem 11, Combining these projectivities, we have The plane dual of Theorem 11 states that the points of contact of a line conic form a point conic. In view of these two theorems and their space duals we now make the following DEFINITION. A conic section or a conic is the figure formed by a point conic and its tangents. A cone is the figure formed by a cone of lines and its tangent planes. The figure formed by a line conic and its points of contact is then likewise a conic as defined above ; i.e. a conic (and also a cone) is a self-dual figure. The duals of Pascal's theorem and its special cases now give us a set of theorems of the same consequence for point conies as for line conies. We content ourselves with restating Brianchon's theorem (Theorem 3') from this point of view. BRIANCHON'S THEOREM. If the sides of a simple hexagon are tan- gents to a conic, the lines joining opposite vertices are concurrent; and conversely. It follows from the preceding discussion that in forming the plane duals of theorems concerning conies, the word conic is left unchanged, while the words point (of a conic) andjJa^$wL(of a conic) are inter- changed. "We shall also, in the future, make use of the phraseTa conic passes through a point P, and P is on the conic, when P is a point of a conic, etc. DEFINITION. If the points of a plane figure are on a conic, the figure is said to be inscribed in the conic; if the lines of a plane figure are tangent to a conic, the figure is said" to be circumscribed about the conic. TA]S T GE^TS 119 EXERCISES 1. State the plane and space duals of the special cases of Pascal's theorem. 2. Construct a conic, given (1) five tangents, (2) four tangents and the point of contact of one of them, (3) three tangents and the points of contact of two of them. 3. ABX is a triangle whose vertices are on a conic, and a, I. x are the tan- gents at A, B, X respectively. If .4, L are given points and X is variable, determine the locus of (1) the center of perspectivity of the triangles ABX and abx ; (2) the axis of perspectivity. JS 4. X, Y, Z are the vertices o a variable triangle, such that X, Y are always on two given lines a, I respectively, while the sides XY, ZX, ZY always pass through three given points P, A, B respectively. Show that the locus of the point Z is a point conic containing A,B,D = (ah), J/ = (AP)l, and JV~ = (BP)a (Maclaurin's theorem). Dualize. (The plane dual of this theorem is known as the theorem of Braiken ridge.) 5. If a simple plane fl-point varies in such avray that its sides always pass through n given points, while n 1 of its vertices are always on n 1 given lines, the nth vertex describes a conic (Poncelet). 6. If the vertices of two triangles are on a conic, the six sides of these two triangles are tangents of a second conic ; and conversely.' Corresponding to every point of the first conic there exists a triangle having this point as a vertex, whose other two vertices are also on the first conic and whose sides are tangents to the second conic. Dualize. 7. If two triangles in the same plane are perspective, the points in \\hich the sides of one triangle meet the nonhomologous sides of the other are on the same conic ; and the lines joining the vertices of one triangle to the non- homologous vertices of the other are tangents to another conic. 8. If A , B, C, D be the vertices of a complete quadrangle, whose sides AB, AC, AD, BC, BD, CD are cut by a line in the points P, Q, R, S, T, V respectively, and if E, F^ff, K, L, JJf are respectively the harmonic conjugates of these points with respect to the pairs of vertices of the quadrangle so that we have H (AB, PE), H (A C, QF), etc., then the six points E, F, G t A", , AT are on a conic which also passes through the diagonal points of the quadrangle (Holgate, Annals of Mathematics, Ser. 1, Vol. Til (1893), p. 73). 9. If a plane a cut the six edges of a tetrahedron in six distinct points, and the harmonic conjugates of each of these points with respect to the two vertices of the tetrahedron that lie on the same edge are determined, then the lines joining the latter six points to any point of the plane a are on a cone, on which are also the lines through and meeting a pair of opposite edges of the tetrahedron (Holgate, Annals of Mathematics, Ser. 1, Tol. VII (1893), p. 73). 10. Given four points of a conic and the tangent at one of them, construct the tangents at the other three points. Dualize. 11. A, A', B, B' are the, vertices of a quadrangle, and m, n are two lines in the plane of thja quadrangle which meet on AA'. M Is a variable point 120 CONIC SECTIONS [CHAP. V OIL m, the lines BM, B'M meet n in the points N, N' respectively ; the lines ANj A'N' meet in a point P. Show that the locus of the lines PJ\1 is a line conic, -which contains the lines m, p = P(n, BB'), and also the lines A A', BB', A'B'y AB (Amodeo, Lezioni di Geometria Projettiva, Naples (1905), p. 331). 12. Use the result of Ex. 11 to give a construction of a line conic deter- mined by five given lines, and show that by means of this construction it is possible to obtain two lines of the conic at the same time (Amodeo, loc. cit.). 13. If a, 5, c are the sides of a triangle whose vertices are on a conic, and m, m' are two lines meeting on the conic which meet a, &, c in the points A, B, C and A', B', C' respectively, and which meet the conic again in N 9 N' respec- tively, we have ABCNj^A'B'C'N' (cf. Ex. 6). 14. If A, B, C, D are points on a conic and a, 5, c, d are the tangents to the conic at these points, the four diagonals of the simple quadrangle ABCD and the simple quadrilateral abed are concurrent. 44. The polar system of a conic. THEOREM 13. If P is a point in THEOREM 13'. If pis a line in the the plane of a conic, lut not on the plane of a conic, but not tangent to conic, the points of intersection of the tangents to the conic at all the pairs of points which are collinear with P are on a line, which also con*- tains the harmonic conjugates of P with respect to these pairs of points. the conic, the lines joining the points of contact of pairs of tangents to the conic which meet on p pass through a point P, through which pass also the harmonic conjugates ofp with respect to these pairs of tangents. R . 58 Proof. Let P v P% and P^, P be two pairs of points on the conic which are collinear with P, and let p v p z be the tangents to the conic at JJ, P 2 respectively (fig. 58). If 2> v D 2 are the points (%%)(%%) an ] POLAR SYSTEM 121 respectively, the line D^D^ passes through the intersection Q of jt\ 3 $ (Theorem 8). Moreover, the point P r in which Z^D* meets 2J2* is the harmonic conjugate of P with respect to P 13 R (Theorem 6, Chap. IT). This shows that the line D^D* QP ! is completely determined by the pair of points P^ P,. Hence the same line QP' is obtained by replacing P 3> P by any other pair of points on the conic collinear with P, and distinct from P v P,. This proves Theorem 13. Theorem 13' is the plane dual of Theorem 13. DEFINITION. The line thus asso- DEFINITION. The point thus ciated with any point P in the associated with any line p in the plane of a conic, but not on the plane of a conic, but not tangent conic, is called the polar of P to the conic, is called thereof p with respect to the conic. If P with respect to the conic. If p is is a point on the conic, the polar a tangent to the conic, the pole is is defined as the tangent at P. defined as the point of contact of P. THEOKEM 14. TJie line joining THEOREM 14'. The point of two diagonal points of any com- intersection of two diagonal lines plete quadrangle whose vertices of any complete quadrilateral are points of a conic is the polar wJiose sides are tangent to a conic of the other diagonal point with is the pole of the other diagonal respect to the conic. line with respect to the conic. Proof. Theorem 14 follows immediately from the proof of Theo- rem 13. Theorem 14' is the plane dual of Theorem 14 THEOREM 15. The polar of a THEOREM 15'. Tlie pole of a point P with respect to a conic line p with respect to a conic is passes through the points of con- on the tangents to the conic at the tact of the tangents to the conic poi/its in which p meets the conic, through P, if such tangents exist, if such paints exist Proof. Let P l be the point of contact of a tangent through. P, and let P%> PS be any pair of distinct points of the conic collinear with P. The line through JJ and the intersection of the tangents at jg, P% meets the line J%P% in the harmonic conjugate of P with, respect to j?fjf (Theorem 9, Cor.). But the line thus determined is the polar of P (Theorem 13). This proves Theorem 15. Theorem 15' is its plane dual THEOREM 16. If p is the polar of a point P with respect to a conic, P is the pole of p with respect to the same conic. 122 COXIC SECTIONS [CHAP.V If P is not on the conic, this follows at once by comparing Theo- rem 13 with Theorem 13'. If P is on the conic, it follows immediately from the definition. THEOREM 17. If the polar of a point P passes through a point Q, the polar of Q passes through P. Proof. If P or Q are on the conic, the theorem is equivalent to Theorem 15. If neither P nor Q is on the conic, let PP^ be a line P 59 meeting the conic in two points, P^ P 2 . If one of the lines P^Q, P 2 Q is a tangent to the conic, the other is also a tangent (Theorem 13); the line P^ = PP is then the polar of Q, which proves the theorem under this hypothesis. If, on the other hand, the lines I$Q,P 2 Q meet the conic again in the points P& P respectively (fig. 59), the point (IJJFJ) (P Z P^ is on the polar of Q (Theorem 14). By Theorems 13 and 14 the polar of (J?-E) (P Z P^) contains the intersection of the tangents at P v R and the point Q. By hypothesis, however, and Theorem 13, the polar of P contains these points also. Hence we have (P^ (P%P^) = P, which proves the theorem. COROLLAEY 1. If two vertices of a triangle are the yoles of their opposite sides with respect to a conic, the third vertex is the pole of its opposite side. DEFINITION. Any point on the polar of a point P is said to be conjugate to P with regard to the conic; and any line on the pole POLAE SYSTEM 123 of a line p is said to be conjugate to p with regard to the conic. The figure obtained from a given figure in the plane of a conic by constructing the polar of every point and the pole of every line of the given figure with regard to the conic is called the polar or polar reciprocal of the given figure with regard to the conic.* A triangle, of which each vertex is the pole of the opposite side, is said to be self -polar or self-conjugate with regard to the conic. COROLLARY 2. TJie diagonal triangle of a complete quadrangle ichose vertices are on a conic, or' of a complete quadrilateral whose sides are tangent to a conic, is self -polar with regard to the co/iic; and, conversely, every self -polar triangle is the diagonal triangle of a complete quad- rangle whose points are on the conic, and of a complete quadrilateral whose sides are tangent to the conic. Corresponding to a given self-polar triangle, one vertex or side of such a quadrangle or quadrilateral may be chosen arbitrarily on the conic. Theorem 17 may also be stated as follows : If P is a variable point on a line q, its polar p is a variable line through the pole Q of q. In the special case where q is a tangent to the conic, we have already seen (Theorem 12) that we have [^AM- If Q is not on q, let A (fig. 60) be a fixed point on the conic, a the tangent at A, JT the point (distinct from A, if AP is not tangent) in which AP meets the conic, and x the tangent at X. "We then have, by Theorem 12, By Theorem 13, (ax) is on p, and hence p = Q (ax). Hence we have [*]*[*] If P ! is the point pq, this gives But since the polar of P ! also passes through P, this projectivity is an involution. The result of this discussion may then be stated as follows : * It was by considering the polar reciprocal of Pascal's theorem that Brianchon derived the theorem named after him. This method was fully developed "by Poncelet and Gergonne in the early part of the last century in connection with the principle of duality. 124 C03STC SECTIONS [CHAP.Y THEOREM 18. On any line not a tangent to a given conic the pairs of conjugate points are pairs of an involution. If the line meets the conic in two points, these points are the doulle points of the involution. COROLLARY. As a point P varies over a pencil of points, its polar with respect to any conic varies over a projectile pencil of lines. 60 . DEFINITION. The pairing of the points and lines of a plane brought about by associating \vith every point its polar and with every line its pole with respect to a given conic in the plane is called a polar system. EXERCISES 1. If in a polar system two points are conjugate to a third point A , the line joining them is the polar of A. 2. State the duals of the last two theorems. ?*' 3. If a and Z> are two nonconjugate lines in a polar system, every point A of a has a conjugate point B on 5. The pencils of points [A] and B] are protective ; they are perspective if and only if a and I intersect on the conic of the polar system. < 4. Let A be a point and 5 a line not the polar of A with respect to a given conic, but in the plane of the conic. If on any line I through A we determine that point P which is conjugate with the point $, the locus of P is a conic passing through A and the pole B of 5, unless the line AB is tangent to the ] POLAB SYSTEM 125 conic, In which case the locus of P is a line. If AB is not tangent to the conic, the locus of P also passes through the points in which b meets the given conic (if such points exist), a-nd also through the points of contact of the tangents to the given conic through A (if such tangents exist). Dualize (Reye-Holgate, Geometry of Position, p. 106). 5. If the vertices of a triangle are on a given conic, any line conjugate to one side meets the other two sides in a pair of conjugate points. Conversely, a line meeting two sides of the triangle in conjugate points passes through the pole of the side (von Staudt). 6. If two lines conjugate with respect to a conic meet the conic in two pairs of points, these pairs are projected from any point on the conic by a harmonic set of lines, and the tangents at these pairs of points meet any tangent in a harmonic set of points. 7. ^Vith a given point not on a given conic as center and the polar of this point as axis, the conic is transformed into itself by a homology of period two. 8. The Pascal line of any simple hexagon whose vertices are on a conic is the polar with respect to the conic of the Brianchon point of the simple hexagon whose sides are the tangents to the conic at the vertices of the first hexagon. 9. If the line joining two points A, B, conjugate with respect to a conic, meets the conic in two points, these two points are harmonic with A 9 B. 10. If in a plane there are given two conies Cf and C|, and the polars oi all the points of Cf with respect to C| are determined, these polars are the tangents of a third conic. 11. If the tangents to a given conic meet a second conic in pairs of points, the tangents at these pairs of points meet on a third conic. 12. Given five points of a conic (or four points and the tangent through one of them, or any one of the other conditions determining a conic), sho^w how to construct the polar of a given point with respect to the conic, 13. If two pairs of opposite sides of a complete quadrangle are pairs ot conjugate lines with respect to a conic, the third pair of opposite sides are conjugate -with respect to the conic (Hesse). 14. If each of two triangles in a plane is the polar of the other with respect to a conic, they are perspective, and the axis of perspectivity is the polar of the center of perspectivity (Chasles). 15. Two triangles that are self -polar with respect to the same conic have their six vertices on a second conic and their six sides tangent to a third conic (Steiner). 16. Regarding the Desargues configuration as composed of a quadrangle and a quadrilateral mutually inscribed (cf. 18, Chap. II), show that the diagonal triangle of the quadrangle is perspective with the diagonal triangle of the quadrilateral. 17. Let A , B be any two conjugate points with respect to a conic, and let the lin^s AM, BM joining them to an arbitrary point of the conic meet the latter again in the points C, D respectively. The lines AD, BC will then meet on the conic, and the lines CD and AB are conjugate. Dualize. 126 CONIC SECTIONS [CHAP.V 45. Degenerate conies. For a variety of reasons it is desirable to regard two coplanar lines or one line (thought of as two coincident lines) as degenerate cases of a point conic; and dually to regard two points or one point (thought of as two coincident points) as degenerate cases of a line conic. This conception makes it possible to leave out the restriction as to the plane of section in Theorem 1. For the section of a cone of lines by a plane through the vertex of the cone consists evidently of two (distinct or coincident) lines, i.e. of a degenerate point conic ; and the section of a cone of planes by a plane through the vertex of the cone is the figure formed by some or all the lines of a flat pencil, i.e. a degenerate line conic. EXERCISE Dualize in all possible ways the degenerate and nondegenerate cases of Theorem 1. Historically, the first definition of a conic section was given by the ancient Greek geometers (e.g. Mensechmus, about 350 B.C.), who defined them as the plane sections of a "right circular cone." In a later chapter we will show that in the " geometry of reals " any nondegenerate point conic is protectively equivalent to a circle, and thus that for the ordinary geometry the modern projective definition given in 41 is equivalent to the old definition. We are here using one of the modern definitions because it can be applied before devel- oping the Euclidean metric geometry. Degenerate conies would be included in our definition (p. 109), if we had not imposed the restriction on the generating projective pencils that they be nonperspective ; for the locus of the point of intersection of pairs of homologous lines in two perspective flat pencils in the same plane consists of the axis of perspectivity and the line joining the centers of the pencils. It will be seen, as we progress, that many theorems regarding non- degenerate conies apply also when the conies are degenerate. For example, Pascal's theorem (Theorem 3) becomes, for the case of a degenerate conic consisting of two distinct lines, the theorem of Pappus already proved as Theorem 21, Chap. IV (cf. in particular the corollary). The polar of a point with regard to a degenerate conic consisting of two lines is the harmonic conjugate of the point with respect to the two lines (cf. the definition, p. 84, Ex. 7). Hence the polar system of a degenerate conic of two lines (and dually of two points) determines an involution at a point (on a line). 45,40] THEOREM OF DBS ARGUES 127 EXERCISES 1. State Brianchon's theorem (Theorem 3') for the case of a degenerate line conic consisting of two points. 2. Examine all the theorems of the preceding sections -with reference to their behavior when the conic in question becomes degenerate. 46. Desargues's theorem on conies. THEOREM 19. If the vertices of a complete quadrangle are oil a conic which meets a line in two points, the latter are a pair in the invo- lution determined on the line ly the pairs of o^iosite sides of the quad/ 'anyk >* Proof. Reverting to the proof of Theorem 2 (fig. 51), let the line meet the conic in the points B v C and let the vertices of the quad- rangle be A v A*, B v C 2 . This quadrangle determines on the line an involution in which S 9 A and T, U are conjugate pairs. But in the proof of Theorem 2 we saw that the quadrangle A^A^BC determines Q^ST, C^U). Hence the two quadrangles determine the same involution on the line, and therefore B 2 , C l are a pair of the involution determined by the quadrangle A^B^Cy Since the quadrangles A^AJS^ and A^BC determine the same involution on the line when the latter is a tangent to the conic, we have as a special case of the above theorem : COROLLARY. If the vertices of a complete quadrangle are on a conic, the pairs of opposite sides meet the tangent at any other point in pairs of an involution of which the point of contact of the tanyent is a double point. The Desargues theorem leads to a slightly different form of statement for the construction of a conic through five given points : On any line through one of the points the complete quadrangle of the other four determine an involution ; the conjugate in this involution of the given point on the line is a sixth point on the conic. As the Desargues theorem is related to the theorem of Pascal, so are certain degenerate cases of the Desargues theorem related to the degenerate cases of the theorem of Pascal (Theorems 6, 7, 8, 9). Thus in fig. 53 we see (by Theorem 6) that the quadrangle BCRP^ deter- mines on the line P^ an involution in which the points P^ P of the conic are one pair, while the points determined by p v P^ and those * First given by Desargues in 1639 ; cf. OEuvres, Paris, Vol. I (1864), p. 188. 128 CONIC SECTIONS [CHAP.V determined by % P^ P^ are two other pairs. This gives the following special case of the theorem of Desargues : THEOBEM 20. If the vertices of a triangle are on a conic, and a line I meets the conic in two points, the latter are a pair of the involution determined on I by the pair of points in which two sides of the triangle meet Z, and the pair in which the third side and the tangent at the opposite vertex meet I In case I is a tangent to the conic, the point of contact is a double point of this involution. In terms of this theorem we may state the construction of a conic through four points and tangent to a line through one of them as follows : On any line through one of the points which is not on the tangent an involution is deter- mined in which the tangent and the line passing through the other two points determine one pair, and the lines joining the point of contact to the other two points determine another pair. The conjugate of the given point on the line in this involution is a point of the conic. A further degenerate case is derived either from Theorem 7 or Theorem 8. In fig. 54 (Theorem 7) let I be the line P 2 P 3 . The quad- rangle ABP^ determines on I an involution in which JJ, P B are one pair, in which the tangents at P l} P determine another pair, and in which the line P^ determines a double point. Hence we have THEOREM 21. If a line I meets a conic in two points and J%, P are any other two points on the conic, the points in which I meets the conic are a pair of an involution through a double point of which passes the line P^P and through a pair of conjugate points of which pass the tangents at P lt P^ If I is tangent to the conic, the point of contact is the second double point of this involution. The construction of the conic corresponding to this theorem may be stated as follows : Given two tangents and their points of contact and one other point of the conic. On any line I through the latter point is determined an involution of which one double point is the intersection with I of the line joining the two points of contact, and of which one pair is the pair of intersections with I of the two tangents. The conjugate in this involution of the given point of the conic on Us a point of the conic. EXERCISE State the duals of the theorems in this section. 47. Pencils and ranges of conies. Order of contact. The theorems of the last section and their plane duals determine the properties of certain systems of conies which we now proceed to discuss briefly. 47] PENCILS AXD RANGES 129 DEFINITION. The set of all conies through the vertices of a complete quadrangle is called a pencil of conies of Type I (fig. 61). Theorem 19 and its plane dual THEOREM 22. Any line (not tlirougli a vertex of the deter- mining quadrangle) is met ly the conies of a pencil of Type I in the pairs of an involution* DEFINITION. The set of all conks tangent to the sides of a complete quadrilateral is called a range of conies of Type I (fig. 62). give at once : THEOREM 22'. The tangents tlirougli any point (not on a side of the determining quadrilateral) to the conies of a range of Type I are the pairs of an involution. FIG. 63 GOBOLLABY. Through a gen- eral^ point in the plane there is one and only one, and tangent to a general line there are two or no conies of a given pencil of Type I. EIG. 64 COBOLLABY. Tangent to a gen- eral line in the plane there is one and only one, and through a gen- eral point there are two or no conies of a gimn range of Type I. * This form of Desargues's theorem is due to Ch. Sturm, Annales de Mathe*ma- tiques, Vol. XVII (1826), p. 180. t The vertices of the quadrangle are regarded as exceptional points. 130 CONIC SECTIONS [CHA.P. V DEFINITION. The set of all conies through the vertices of a triangle and tangent to a fixed line through one vertex is called a pencil of conies of Type II (fig. 63). DEFINITION. The set of all conies tangent to the sides of a triangle and passing through a fixed point on one side is called a range of conies of Type II (fig. 64). Theorem 20 and its plane dual then give at once : THEOREM 23. Any line in the plane of a pencil of conies of Type II (which does not pass through a vertex of the determin- ing triangle) is met by the conies of the pencil in the pairs of an involution* COROLLARY. Through a general point in the plane there is one and only one conic of the pencil; and tangent to a general line in the plane there are two or no conies of the pencil. THEOREM 23'. The tangents through any point in the plane of a range of conies of Type II (which is not on a side of the determining triangle) to the conies of the range are the pairs of an involution. COROLLARY. Tangent to a gen- eral line in the plane there is one and only one conic of the range; and through a general point in the plane there are two or no conies of the range. DEFINITION. The set of all conies through two given points and tangent to two given lines through these points respectively is called a pencil or range of conies of Type IV* (fig. 65). Theorem 21 now gives at once: THEOREM 24. Any line in the plane of a pencil of conies of Type IV (which does not pass through either of the points common to all the conies of the pencil) is met ly the conies of the pencil in the pairs of an involution. Through any point in the plane (not on either of the lines that are tangent to all the conies of the pencil) the tangents to the conies of the pencil are the pairs of an involution. Tlie line joining the two points common to all the conies of the pencil meets * The classification of pencils and ranges of conies into types corresponds to the classification of the corresponding plane collineations (cf. Exs. 2, 4, 7, below). FIG. Co 47] PENCILS AND RANGES 131 any line in a double point of the involution determined on that line. And {lie point of intersection of the common tangents is joined to any point ly a double line of the involution determined at that point COROLLAEY. Throitgli any general point or tanyent to any general line in the plane there is one and only one conic of the pencil. EXERCISES 1. What are the degenerate conies of a pencil or range of Type 7? The diagonal triangle of the fundamental quadrangle (quadrilateral) of the pencil (range) is the only triangle which is self -polar with respect to two conies of the pencil (range). 2. Let A' 2 and B 2 be any two conies of a pencil of Type /, and let P be any point in the plane of the pencil. If p is the polar of P with respect to .-I 2 , and P' is the pole of p with respect to B' 2 , the correspondence thus established between [P] and [P'] is a projective collineation of Type J, whose invariant triangle is the diagonal triangle of the fundamental quadrangle. The set of all projective collineations thus determined by a pencil of conies of Type I form a group. Dualize. 3. What are the degenerate conies of a pencil or range of Type //? 4. Let a pencil of conies of Type // be determined by a triangle ABC and a tangent a through A. Further, let ' be the harmonic conjugate of a with respect to AB and AC, and let A' be the intersection of a and BC. Then A, a and A', a' are pole and polar with respect to every conic of the pencil ; and no pair of conies of the pencil have the same polars with regard to any other points than A and A'. Dualize, and show that all the collineations determined as in Ex. 2 are in this case of Type //. 5. What are the degenerate conies of a pencil or range of Type JT? 6. Show that any point on the line joining the two points common to all the conies of a pencil of Type IV has the same polar with respect to all the conies of the pencil, and that these all pass through the point of intersection of the two common, tangents. 7. Show that the collineations determined by a pencil of Type IV by the method of Ex. 2 are all homologies (i.e. of Type IV). * The pencils and ranges of conies thus far considered have in com- mon the properties (1) that the pencil (range) is completely defined as soon as two conies of the pencil (range) are given ; (2) the conies of the pencil (range) determine an involution on any line (point) in the plane (with the exception of the lines (points) on the determining points (lines) of the pencil (range)). Three other systems of conies may be defined which likewise have these properties. These new systems * The remainder of this section may be omitted on a first reading. 132 COtflC SECTIONS [CHAP. V may be regarded as degenerate cases of the pencils and ranges already defined. Their existence is established by the theorems given below, which, together with their corollaries, may be regarded as degenerate cases of the theorem of Desargues. "We shall need the following LEMMA. Any conic is transformed lij a protective collineation in the plane of the conic into a conic sucli that the tangents at homologous points are homologous. Proof. This follows almost directly from the definition of a conic. Two projective flat pencils are transformed by a projective collineation into two projective flat pencils. The intersections of pairs of homologous lines of one pencil are therefore transformed into the intersections of the corresponding pairs of homologous lines of the transformed pencils. If any line meets the first conic in a point P, the transformed line will meet the transformed conic in the point homologous with P. Therefore a tangent at a point of the first conic must be transformed into the tangent at the corresponding point of the second conic. THEOREM 25. If a line p Q is a tangent to a conic A 2 at a 'point P OJ and Q is any point of A 2 , then through any point on the plane of A 2 but not on A 2 or p , there is one and only one conic B* through PS and Q, tangent to Pv and such that there is no point except P^ which has the same polar with regard to loth A* and B\ Proof. If P J is any point of the plane not on p Q or 'A 2 , let P be the second point in which P Q P ! meets A* (fig. 66). There is one and only one elation with center P Q and axis P Q Q changing P into P ; (Theorem 9, Chap. III). This elation (by the lemma above) changes A* into another conic B* through the points P^ and Q and tangent to jp . The lines through J^ are unchanged by the elation, whereas their poles (on p ) are subjected to a parabolic projectivity. Hence no point on p Q (distinct from P Q ) has the same polar with regard to A* as with regard to B*. Since A 2 is transformed into B* by an elation, the two conies can have no other points in common than P Q and Q. 47] PEXCILS AXB EAXGES 133 That there is only one eonic B' 2 through P r satisfying the con- ditions of the theorem is to be seen as follows : Let QP meet p in 3, and QP 1 meet p in S r (fig. 66;. The points S and S r must have the same polar with regard to A 2 and any conic J5 2 , since this polar must be the harmonic conjugate of p^ with regard to P Q Q and P Q P. Let p be the tangent to A 2 at P and p f be the tangent to If 2 at P f , and let p and p r meet p Q in I and 2 17 respectively. The points FIG. 67 T and T J have the same polar, namely P Q P, with regard to A 2 and any conic B*. By the conditions of the theorem the projectivity must be parabolic. Hence, by Theorem 23, Cor.,, Chap. IV, Q(P ST,P Q T f S r ). Hence p and p f must meet on P^Q in a point B so as to form the quad- rangle EQPP 1 . This determines the elements P Q , Q, P f > p Q , p f of B*> and hence there is only one possible conic J5 2 . COROLLARY 1. The conies A 2 and 3 s can have no other points in common than P and Q. COROLLARY 2. Any line I not on P^ or Q which meets A 2 and B* meets them in pairs of an involution in which the points of intersection of I with P Q Q <md p Q are conjugate. Proof. Let I meet Jf in N and N v J5 2 in L and L v J%Q in M, and j? in M 1 (fig. 67). Let K and ^ be the points of A 2 which are trans- formed by the elation into L and L^ respectively. By the definition of an elation K and S t are colHnear with Jf, while K is on the line JJ and J^ on LJ%. Let KN^ meet p in jK, and j^JJ meet JCS^ in & 134 COXIC SECTIONS [CHAP. V Then, since JN", E, X v K are on the conic to which. p Q is tangent at ^, we have, by Theorem 6, applied to the degenerate hexagon P^K^EN^N, that S 9 L v and E are collinear. Hence the complete quadrilateral Sit, JEYy JOTp I has pairs of opposite vertices on P Q 3I and P Q M V P Q N and J^Y r JJ and P L r Hence ox . The set of all conies through a point Q and tangent to a line p^ at a point -Z^ and such that no point of p except % has the same polar with regard to two conies of the set, is called a pencil of conies of Type III (tig. 68). DEFINITION. The set of all conies tangent to a line q and tangent to a line p Q at a point P Q , and such that no line on P except p has the same pole with regard to two conies of the set, is called a range of conies of Type III (fig. 69). FIG. 08 FIG. 09 Two conies of such a pencil (range) are said to have contact of the second order, or to osculate, at JJ. Corollary 2 of Theorem 25 now gives at once: THEOREM 26. Any line in the plane of n peticil of conies of Type III, which is not on either of the common points of the pencil, is met hy the conies of the pencil in the pa i/'s of a n iii i*oht t ion. TJi ro ugh any point iti the plane except the ctnnitwn points there is one and only one conic of th e pencil; and tanyent to any line not through either of the common points there are ttro or no w flics of the pencil. THEOREM 26'. Tlirougli any point in tlie plane of a range of conies of Type III, which is not on either of the common tangents of the range, the tangents to the conies of the pen- cil are the pairs of an involution. Tangent to any line in the plane ex- cept the common tangents there is one and only one conic of the range; and through any point not on either of the common tangents there are two or no conies of the range. * This argument has implicitly proved that three pairs of points of a conic, as Nii P Q> such that the lines joining them meet in a point M, are projected from any point of the conic by a quadrangular set of lines (Theorem 16, Chap. VUI). 47] PENCILS AND BAXGES 135 The pencil is determined by The range is determined by the two common points, the com- the two common tangents, the mon tangent, and one conic of the common point, and one conic of pencil. the range. EXERCISES 1. What are the degenerate conies of this pencil and range? 2. Show that the collineation obtained by making correspond to any point P the point P' which has the same polar p with regard to one given conic of the pencil (range) that P has with regard to another given conic of the pencil (range) is of Type 1/7. THEOREM 27. If a line p^ is tangent to a conic A 2 at a point P Qt there is one and only one conic tangent to p$ at P^ and passing through any other point P 1 of the plane of A 2 not on p Q or A 2 which determines for every point of p Q the same polar line as does A~. Proof. Let P be the second point in which P^P* meets A 2 (fig. 70). There is one and only one elation of which 2J is center and p Q axis, changing P to P 1 . This elation changes A 2 into a conic B* through B* P r , and is such that if q is any tangent to A 2 at a point Q, then q is transformed to a tangent $ f of B* passing through gjp , and Q is trans- formed into the point of contact Q r of q r , collinear with Q and JJ. Hence there is one conic of the required type through P f . That there is only one is evident, because if I is any line through JP', any conic B* must pass through the fourth harmonic of P J with regard to lp Q and the polar of lp a as to A* (Theorem 13). By considering two lines I we thus determine enough points to fix B\ COBOLLAEY 1. By duality there is one and only one conic B z tangent to any line not passing through P y 136 CONIC SECTIONS [CHAF.V COROLLARY 2. Any line I not on P^ which meets A* and B* meets them in pairs of an involution one doitble point of which is lp Q) and the other the point of I conjugate to lp Q with respect to A 2 . A dual statement holds for any point L not on p Q . COROLLARY 3. The conies A 2 and B' 2 can have no other point in commoti thati J?J cnid no other tangent iti common than p Q . Proof. If they had one other point P in common, they would have in common the conjugate of P in the involution determined on any line through P according to Corollary 2. DEFINITION. The set of all conies tangent to a given line p Q at a given point P , and such that each point on p has the same polar with regard to all conies of the set, is called a pencil or range of conies of Type V. Two conies of such a pencil are said to have contact of the third order, or to hyperosculate at JJ. Theorem 27 and its first two corollaries now give at once: THEOREM 28. Any line I not on the common point of a pencil of Type V is met ly the conies of the pencil in pairs of an involution one double point of which is the intersection of I with the common tangent. Through any point L not on the common tangent the pairs of tangents to the conies of the pencil form an involution one double line of which is the line joining L to the common point. There is one conic of the set through each point of the plane not on the common tangent, and one conic tangent to each line not on the common point. The pencil or range is determined by the common point, the common tangent, and one conic of the set EXERCISES 1. What are the degenerate conies of a pencil of Type F? 2. Show that the collineation obtained by making correspond to any point P the point Q which has the same pole p with regard to one conic of a pencil of Type V that P has with regard to another conic of the pencil is an elation. 3. The lines polar to a point A with regard to all the conies of a pencil of any of the five types pass through a point A'. The points A and A' are double points of the involution determined by the pencil on the line A A'. Construct A'. Dualize. Derive a theorem on the complete quadrangle as a special case of this one. 4. Construct the polar line of a point A with regard to a conic C 2 being given four points of C* and a conjugate of A with regard to <7 2 . 47] PENCILS AXD BADGES 137 5. Given an involution I on a line Z, a pair of points A and A' on I not conjugate in I, and any other point B on L construct a point B' such that .4 and A' and B and 5' are pairs of an involution I' whose double points are a pair in I. The involution I' may also be described as one which is commu- tative with I, or such that the product of I and T is an involution. 6. There is one and only one conic through three points and having a given point P and line p as pole and polar. 7. The conies through three points and having a given pair of points as conjugate points form a pencil of conies. MISCELLANEOUS EXERCISES 1. If and o are pole and polar with regard to a conic, and *1 and B are two points of the conic collinear with 6>, then the conic is generated by the two pencils A [P] and B [P'] where P and P' are paired in the involution on o of conjugates with regard to the conic. / 2. Given a complete five-point ABCDE. The locus of all points X such that X (BCDE) T A (BCDE) is a conic. A V 3. Given two projective nonperspective pencils, [p] and [<?]. Every line I upon which the projectivity Z[^] -r-l^q] is involutorie passes through a fixed point 0. The point is the pole of the line joining the centers of the pencils with respect to the conic generated by them. ^ 4. If two complete quadrangles have the same diagonal points, their eight vertices lie on a conic (Cremona, Projective Geometry (Oxford, 1885), Chap.XX). 5. If two conies intersect in four points, the eight tangents to them at these points are on the same line conic. Dualize and extend to the cases where the conies are in pencils of Types II-V. 6. All conies with respect to which a given triangle is self -conjugate, and which pass through a fixed point, also pass through three other fixed points. Dualize. 7. Construct a conic through two given points and with a given self- conjugate triangle. Dualize. 8. If the sides of a triangle are tangent to a conic, the lines joining two of its vertices to any point conjugate with regard to the conic to the third vertex are conjugate with regard to the conic. Dualize. 9. If two points P and Q on a conic are joined to two Conjugate points P f 9 Q? on a line conjugate to PQ, then PP' and QQf meet on the conic. 10. If a simple quadrilateral is circumscribed to a conic, and if I is any transversal through the intersection of its diagonals, / will meet the conic and the pairs of opposite sides in conjugate pairs of an involution. Dualize. 11. Given a conic and three fixed collinear points A, B, C. There is a fourth point D on the line AB such that if three sides of a simple quadrangle in- scribed in the conic pass through A, B, and C respectively, the fourth passes, through D (Cremona, Chap. XVII). 138 CONIC SECTIONS [CHAP.V 12. If a variable simple n-line ( even) is inscribed in a conic in such a way that n 1 of its sides pass through n 1 fixed collinear points, then the other side passes through another fixed point of the same line. Dualize this theorem. 13. If two conies intersect in two points A. B (or are tangent at a point A) and two lines through A and B respectively (or through the point of contact .4) meet the conies again in 0, (/ and L, Z/, then the lines OL and O'L' meet on the line joining the remaining points of intersection (if existent) of the two conies. 14. If a conic C 2 passes through the vertices of a triangle which is self- polar with respect to another conic K 2 , there is a triangle inscribed in C 2 and self -polar with regard to J 2 , and having one vertex at any point of C 2 . The lines which cut C 3 and K* in two pairs of points which are harmonically con- jugate to one another constitute a line conic C|, which is the polar reciprocal of C 2 with regard to K* (Cremona, Chap. XXII). 15- If a variable triangle is such that two of its sides pass respectively through two fixed points 0' and lying on a given conic, and the vertices oppo- site them lie respectively on two fixed lines u and u', while the third vertex lies always on the given conic, then the third side touches a fixed conic, which touches the lines u and u'. Dualize (Cremona, Chap. XXII). 16. If P is a variable point on a conic containing A, B, C, and Ms a vari- able line through P such that all throws T (PA, PB: PC, /) are projective, then all lines I meet in a point of the conic (Schroter, Journal fur die reine und angewandte Mathematik, Tol. LXII, p. 222). 17. Given a fixed conic and a fixed line, and three fixed points A, B, C on the conic, let P be a variable point on the conic and let PA, PB, PC meet the fixed line in A', B'* C'. If is a fixed point of the plane and (OA', PB'} = K and (j&TC") = /, then K describes a conic and I a pencil of lines whose center is on the conic described by K (Schroter, loc. cit.). 18. Two triangles ABC and PQR are perspective in four ways. Show that if ABC and the point P are fixed and Q, R are variable, the locus of each of the latter points is a conic (cf. Ex. 8, p. 105, and Schroter, Mathematische Annalen, Vol. II (1870), p. 553). 19. Given six points on a conic. By taking these in all possible orders 60 different simple hexagons inscribed in. the conic are obtained. Each of these simple hexagons gives rise to a Pascal line. The figure thus associated with any six points of a conic is called the hexagramma mysticum.* Prove the following properties of the hexagramma niysticum : i. The Pascal lines of the three hexagons PiPoPgP^Pg, P^P^P Z P^P 5 P^ and PiPgPsP^PgP^ are concurrent. The point thus associated with such a set of three hexagons is called a Steiner point. ii. There are in all 20 Steiner points. * On the Pascal hexagram cf . Steiner-Schroter, Vorlesungen tiber Synthetische Geometrie, Vol. II, 28 ; Salmon, Conic Sections in the Notes ; Christine Ladd, American Journal of Mathematics, Vol. IE (1879), p. 1. 47] EXERCISES 139 iii. From a given simple hexagon five others are obtained by permuting in all possible ways a set of three vertices no two of which are adjacent. The Pascal lines of these six hexagons pass through two Steiner points, which are called conjugate Steiner points. The 20 Steiner points fall into ten pairs of conjugates. iv. The 20 Steiner points lie by fours on 15 lines called Steiner lines. v. AVhat is the symbol of the configuration composed of the 20 Steiner points and the 15 Steiner lines ? 20. Discuss the problem corresponding to that of Ex. 19 for all the special cases of Pascal's theorem. 21. State the duals of the last two exercises. 22. If in a plane there are given two conies, any point A has a polar with respect to each of them. If these polars intersect in A', the points A , A' are conjugate with respect to both conies. The polars of A / likewise meet in A. In this way every point in the plane is paired with a uniqxte other point. By the dual process every line in the plane is paired with a unique line to which it is conjugate with respect to both conies. Show that in this correspondence the points of a line correspond in general to the points of a conic. All such conies which correspond to lines of the plane have in common a set of at most three points. The polars of every such common point coincide, so that to each of them is made to correspond all the points of a line. They form the excep- tional elements of the correspondence. Dualize (Reye-Holgate, p. 110).* 23. If in the last exercise the two given conies pass through the vertices of the same quadrangle, the diagonal points of this quadrangle are the " common points " mentioned in the preceding exercise (Reye-Holgate, p. 110). 24. Given a cone of lines with vertex and a line u through 0. Then a one-to-one correspondence may be established among the lines through by associating with every such line a its conjugate a' with respect to the cone lying in the plane au. If, then, a describes a plane ?r, of will describe a cone of lines passing through u and through the polar line of ?r, and which has in common with the given cone any lines common to it and to the given cone and the polar plane of u (Reye-Holgate, p. 111).* 25. Two conies are determined by the two sets of five points A, B, C,D,E and A,B,C 9 H, K. Construct the fourth point of intersection of the two conies (Castelnuovo, Lezioni di Geometria, p. 391). 26. Apply the result of the preceding Exercise to construct the point P such that the set of lines P(A, JB, C, D, E) joining P to the vertices of any given complete plane five-point be projective with any given set of five points on a line (Castelnuovo, loc. cit.). 27. Given any plane quadrilateral, construct a line which meets the sides of the quadrilateral in a set of four points projective with any given set of four collinear points. * The correspondences defined in Exs. 22 and 24 are examples of so-called quadratic correspondences. 140 COKIC SECTIONS [CHAP.V 28. Two sets of five points .1, B, C, Z>, E and .4, B, H, K 9 L determine two conies which, intersect again in two points A", I". Construct the line XT and show that the points JY", F are the double points of a certain involution (Castelnuovo, loc. cit.). 29. If three conies pass through two given points A, B and the three pairs of conies cut again in three pairs of points, show that the three lines joining these pairs of points are concurrent (Castelnuovo, loc. cit.). 30. Prove the converse of the second theorem of Desargues : The conies passing through three fixed points and meeting a given line in the pairs of an involution pass through a fourth fixed point. This theorem may be used to construct a conic, given three of its points and a pair of points conjugate with respect to the conic. Dualize (Castelnuovo, loc. cit.). 31. The poles of a line with respect to all the conies of a pencil of conies of Type 1 are on a conic which passes through the diagonal points of the quadrangle defining the pencil. This conic cuts the given line in the points in which the latter is tangent to conies of the pencil. Dualize. 32. Let p be the polar of a point P with regard to a triangle ABC. If P varies on a conic which passes through ^i , B, C* then p passes through a fixed point Q (Cayley, Collected Works, Vol. I, p. 301). 33. If two conies are inscribed in a triangle, the six points of contact are on a third conic. CHAPTER VI ALGEBRA OF POINTS AND OUTE -DIMENSIONAL COORDINATE SYSTEMS 48, Addition of points. That analytic methods may be introduced into geometry on a strictly protective basis was first shown by von Staudt.* The point algebra on a line which is defined in this chapter without the use of any further assumptions than A, E, P is essentially equivalent to von Staudt's algebra of throws (p. 60), a brief account of which will be found in 55. The original method of von Staudt has, however, been considerably clarified and simplified by modern researches on the foundations of geometry.f All the definitions and theorems of this chapter before Theorem 6 are independent of As- sumption P. Indeed, if desired, this part of the chapter may be read before taking up Chap. IV. Given a line I, and on I three distinct (arbitrary) fixed points which for convenience and suggestiveness we denote by P , P^ P n , we define two one-valued operations $ on pairs of points of I with reference to the fundamental points P Q) JJ, P^. The fundamental points are said to determine a scale on I. DEFINITION. In any plane through I let L and IL be any two lines through P^, and let Z be any line through 1% meeting / and II in points A and A f respectively (fig. 71). Let P x and P y be any two points of I, and let the lines P X A and P y A* meet II and Z in the points X and Y respectively. The point P x+y , in which the line XT meets Z, is called the sum of the points P x and P v (in symbols J^ + ^ = -^ +y ) in * K G. C. von Staudt (1798-1867), BeitrSge zur Geometric der Lage, Heft 2 (1857), pp. 166 et seq. This "book is concerned also with the related question of the inter- pretation of imaginary elements in geometry. t Of., for example, G. Hessenberg, Ueber einen Geometrischen Calcul, Acta Mathematiea, Vol. XXIX, p. 1. J By a one-valued operation o on a pair of points A, B is meant any process whereby with every pair A^ B is associated a point O, which is unique provided the order of A, B is given; in symbols AoBC. Here "order" has no geo- metrical significance, but implies merely the formal difference of AoB and Bo A. If AoB BoA, the operation is commutative; if (AoB)oC=Ao(BoC), the opera- tion is associative. 141 142 ALGEBEA OF POINTS [CHAP. VI the scale ^, P v P n . The operation of obtaining the sum of two points is called addition* THEOREM 1. If P x and P y are distinct from P Q and P ee) Q(P a ,P x P Q9 y ) is a necessary and sufficient condition for the equality This follows immediately from the definition, AXA!Y being a quadrangle which determines the given quadrangular set. COROLLARY 1. If P x is any point of I, we have P x +P Q = P + P x =iI x> This is also an immediate consequence of the definition. COROLLARY 2. The operation of addition is one-valued for every pair of points P& P y of I, except for the pair P mi Jf^. (A, E) This follows from the theo- rem above and the corollary of * The historical origin of this con- struction will he evident on inspection of the attached figure. This is the figure which results, if we choose for & the "line at infinity" in the plane in the sense of ordinary Euclidean geometry (cf. p. 8). The construction is clearly equivalent to a translation of the vector P Q P y along the line Z, which brings its initial point into coincidence with the terminal point of the vector PoP x , which is the ordinary construction for the sum of two vectors on a line. 48] ADDITION 143 Theorem 3, Chap. II, in case P x and P y are distinct from JfJ and P um If one of the points P x , P y coincides with P Q or P 9 , it follows from Corollary 1. COROLLARY 3. The operation of addition is associative; i.e. for any three points P^ P v , P s for which the above expressions are defined. (A, E) Proof (fig. 73). Let J^-f J^ be determined as in the definition by- means of three lines ?, 7, 7 and the line XF. Let the line ^Y be denoted by Z ', and by means of l*JL 9 7 ' construct the point (J + P y ) 4- ^, . 73 which is determined by the line XZ, say. If now the point P y -}~P z be constructed by means of the lines l m , l, 7 ; , and then the point P x + (P y + %) be constructed by means of the lines l m , l! l^ it will be seen that the latter point is determined by the same line XZ. COROLLARY 4 The operation of addition is commutative; i.e. for every pair of points P x , P y for which the operation is defined. (A, E) Proof. By reference to the complete quadrangle AXA ! Y (fig. 71) there appears the quadrangular set Q(P m P y P , 1J, jyj +y ), which by the theorem implies that P y +I^^P x ^ jr But, by definition, P x +P y = P x + y . Hence 144 ALGEBBA. OF PODTTS [CHAP.TI THEOREM 2. Any three points P x , 1^, P a (P a = R) satisfy the relation P ppp p p p p J -<x> J - J - J -i - L 'o- t - c ' t i.e. the correspondence established by making each point P x of I corre- spond to P x r = P x +P ai where P a (= P*) is any fixed point of I, is protective. (A, E) Proof. The definition of addition (fig. 71) gives this projectivity as the result of two perspectivities:* The set of all projectivities determined by all possible choices of P a in the formula Pi-P^P^ is the group described in Example 2, p. 70. The sum of two points P a and P b might indeed have been defined as the point into which P b is transformed when P is transformed into P a by a projectivity of this group. The associative law for addition would thus appear as a special case of the associative law which holds for the composition of correspondences in general ; and the commutative law for addition would be a consequence of the commutativity of this particular group. FIG. 74 49. MtdtipUcation of points. DEFIXITIOX. In any plane through I let ? , l v k be any three lines through P Q9 P v P^ respectively, and let ^ meet 4" and I* in points A and B respectively (fig. 74). Let P x) P y be any two points of 1 9 and let the lines P X A and P y B meet 1 9 and Z in the points X and T respectively. The point 2^ in which the line XY meets I is * To make fig. 71 correspond to the notation of this theorem, P must be identified with P a . 49] MULTIPLICATION 145 called the product ofP x by P y (in symbols P x -P y =P^)m the scale P^ P^ P x on /. The operation of obtaining the product of two points is called multiplication* Each of the points P x , P u is called a factor ' of the product P x - P r THEOREM 3. If P x and P v are any tv:o i^ints of I distinct from P Q , P l} P X) Q(P Q P x P ly P*P y P xlf ) is necessary and sufficient for the equality P X -^P^. (A,E) This follows at once from the definition, AXJBY being the defining quadrangle. COROLLARY 1. For any point P, (=F P^) on I ice Jtrn-e the relations P^P = P X P, = P X ; P^P^P^P^P,; P X P^P^P^P^ (A,E) This follows at once from the definition. COROLLARY 2. TJie operation of multiplication is one-valued for every pair of points P x , P y of I, except P^ - P^ and P^ P^. (A, E) This follows from Corollary 1, if one of the points P 3) P y coincides with %, P v or P n . Otherwise, it follows from the corollary, p. 50, in connection with the above theorem. Po * The origin of this construction may also "be seen in a simple construction of metric Euclidean geometry, which results from the construction of the definition "by letting the line Z he the "line at infinity" (cf. p. 8). In the attached figure which gives this metric construction we have readily, from similar triangles, the proportions: j^ = y^ = JiP , ^ PO JPy PO y Po-Pary which, on taking the segment P Pi= 1, gives the desired result PoPy= PoP 146 ALGEBBA OF POINTS [CHAP. VI COBOLLARY 3. The operation of multiplication is associative; i.e. we ha re (^ - P y ) P z = P A - (P y P z ) for every three points P x , JJ, P z for which these products are defined. (A, E) Proof (fig. 76). The proof is entirely analogous to the proof for the associative law for addition. Let the point P x P y be constructed lo FIG. 70 as in the definition by means of three fundamental lines 1 Q) 1 13 the point P xy being determined by the line XT. Denote the line . by l[, and construct the point P^ P z = (P x P y ) *P S) using the lines Z , l[, / as fundamental. Further, let the point P y P s = ^ be constructed by means of the lines Z , V v ?, and then let P x *I> yz = P x * ( 1J) be con- structed by means of Z , I v I*. It is then seen that the points P x P us and P^ JP are determined by the same line. By analogy with Theorem 1, Cor. 4, we should now prove that mul- tiplication is also commutative. It will, however, appear presently that the commutativity of multiplication cannot be proved without the use of Assumption P (or its equivalent). It must indeed be clearly noted at this point that the definition of multiplication requires the first factor P x in a product to form with JjJ and P l a point triple of the quadrangular set on I (cf. p. 49); the construction of the product is therefore not independent of the order of the factors. Moreover, the fact that in Theorem 3, Chap. II, the quadrangles giving the points of the set are similarly placed, was essential in the proof of that 49] MULTIPLICATION 147 theorem. "We cannot therefore use this theorem to prove the com- mutative law for multiplication as in the case of addition. An important theorem analogous to Theorem 2 is, however, inde- pendent of Assumption P. It is as follows : THEOREM 4 If the relation P x -P >f = P xtJ holds Ittvecn any three 'points P^ P v , P jy on I distinct from f^ we Jut re P^P^P^ ^P^P^ and also .SJJJfJJJ P^P^P^; i.e. the correspondence established "by makintj each point P x of I correspond to P^P x *P a (or to P x '~P a -Pj}, where P a is a ny fixed point of I distitict from zero, is projective. (A, E) Proof. The definition of multiplication gives the first of the above projectivities as the result of two perspectivities (fig. 76) : The second one is obtained similarly. In fig. 76 we have The set of all projectivities determined by all choices of P a in the for- mula Px = P x -P a is the group described in Example 1, p. 69. The proper- ties of multiplication may be regarded as properties of that group in the same svay that the properties of addition arise from the group described in Example 2, p. 70. In particular, this furnishes a second proof of the associative law for multiplication. THEOREM 5. Multiplication is distributive with respect to addition; i.e. if P x) P y , P z are any three points on I (for which the operations "below are defined), we have P z .(P x +P y )=P s .P I +P,.P y , a nd(Z+P l ).P z =P I .P s + P v .P,. (A,E) Proof. Place P x +P y = P x + y , P Z -P^P^ P s -P y = P, y , and P z -P^ = P^+ yy By Theorem 4 we then have 5 -? ^l-VV^c -i- y "A * ^0 -? *zx *zy2z(x + y)' But by Theorem 1 we also have (&%%> %>%%+) Hence, by Theorem 1, Cor., Chap. IV, we have Q^J^JJ, P^P^^ which, by Theorem 1, implies -^ B -r-^=-^ (a:+ y ) . The relation is proved similarly. 148 ALQEBEA OF POINTS [CHAP.VI 50. Tlie commutative law for multiplication. With the aid of Assumption P we will now derive finally the commutative law for multiplication : THEOREM 6. TJie operation of multiplication is commutative; i.e. we hare P x JJ = P y - P for every pair of points P x , P y of I for which these two products are defined. (A., E, P) Proof. Let us place as "before P x - P y = P xy , and J - P x = P yc . Then, by the first relation of Theorem 4, and interchanging the points P^ P^ we have and from the second relation of the same theorem we have By Theorem 17, Chap. IV, this requires P^ P^. In view of the fact already noted, that the fundamental theorem of protective geometry (Theorem 17, Chap. IT) is equivalent to Assumption P, the proof just given shows further: THEOKEM 7. Assumption P is necessary and sufficient for the com- mutative law for multiplication* (A, E) 51. The inverse operations. DEFINITION. Given two points P a , J% on Z, the operation determining a point JP satisfying the relation P^ + P x =iI is called subtraction; in symbols IP a =zP x . The point P x is called the difference of J? from P a . Subtraction is the inverse of addition. The construction for addition may readily be reversed to give a con- struction for subtraction. The preceding theorems on addition then give : THEOEEM 8. Subtraction is a one-valued operation for every pair of points P a > P b on I, except the pair P^ j. (A, E) COROLLAEY. We have in particular P a P a P Q for every point P a (^PJonl (A, E) * The existence of algebras in which multiplication is not commutative is then sufficient to establish the fact that Assumption P is independent of the previous Assumptions A and E. For in order to construct a system (cf. p. 6) which satisfies Assumptions A and E without satisfying Assumption P, we need only construct an analytic geometry of three dimensions (as described in a later chapter) and use as a basis a noncommutative number system, e.g. the system of quaternions. That the fundamental theorem of projective geometry is equivalent to the commutative law for multiplication was first established by Hilbert, who, in his Foundations of Geometry, showed that the commutative law is equivalent to the theorem of Pappus (Theorem 21, Chap. IV). The latter is easily seen to be equivalent to the funda- mental theorem. 60,31,32] ABSTRACT XU^IBEE SYSTEM 149 DEFINITION. Given two points P a , P b on / ; the point P x determined by the relation J^-^ = ^ is called the quotient of J% by P a (also the rutio of P b to P a ) ; in symbols JIJ/JJ = Jf, or J : P a = JJ. The operation deter- mining j/2 is called diinsio/i; it is the inverse of multiplication.* The construction for multiplication may also be reversed to give a construction for division. The preceding theorems on multiplication then give readily : THEOREM 9. Division is a one-valued operation for every pair of points P a , P b on I except the pairs P Q , P Q and .%, -%. (A, E) COKOLLAHY. We hare in particular PJ% = P^ ./_ = J?, ^/J = , etc., for every poi lit P a o>i I distinct from JjJ and 1^. (A, E) Addition, subtraction, multiplication, and division are known as the four rational operations. 52. The abstract concept of a number system. Isomorphism. The relation of the foregoing discussion of the algebra of points on a line to the foundations of analysis must now -be briefly considered. "With the aid of the notion of a group (cf. Chap. Ill, p. 66), the general con- cept of a number system is described simply as follows : DEFINITION. A set N of elements is said to form a number system, provided two distinct operations, which we will denote by and respectively, exist and operate on pairs of elements of N under the following conditions: 1. The set N forms a group with respect to e. 2. The set N forms a group with respect to o, except that if i + is the identity element of N with respect to e, no inverse with respect to o esists for i + .f If a is any element of N, a i+ = ^0 a = i+. 3. Any three elements a, &, c of N satisfy the relations a (b c) = (5)(ac) and (l@c)ea=(bQa)e(cd). The elements of a number system are called numbers; the two oper- ations and are called addition and mult implication respectively. If a number system forms commutative groups with respect to both addition and multiplication, the numbers are said to form * What we have defined is more precisely right-handed division. The left-handed quotient is defined similarly as the point P x determined by the relation P x P a P&, In a commutative algebra they are of course equivalent. t The identity element i+ in a number system is usually denoted by (zero). $ The class of all ordinary rational numbers forms a field; also the class of real numbers; and the class of all integers reduced modulo p (p a prime), etc. 150 ALGEBRA OF POINTS [CHAP.YI On the basis of this definition may be developed all the theory relating to the rational operations i.e. addition, multiplication, sub- traction, and division in a number system. The ordinary algebra of the rational operations applying to the set of ordinary rational or ordinary real or complex numbers is a special case of such a theory. TJie whole terminology of tins algebra, in so far as it is definable in terms of the four rational operations, will in the future be assumed as defined. We shall not, therefore, stop to define such terms as reciprocal of a number, exponent, equation, satisfy, solution, root, etc. The element of a number system represented by a letter as a will be spoken of as the value of a. A letter which represents any one of a set of numbers is called a variable; variables will usually be denoted by the last letters of the alphabet. Before applying the general definition above to our algebra of points on a line, it is desirable to introduce the notion of the abstract equivalence or isomorphism between two number systems. ox^ If two number systems are such that a one-to-one reciprocal correspondence exists between the numbers of the two systems, such that to the sum of any two numbers of one system there corresponds the sum of the two corresponding numbers of the other system; and that to the product of any two numbers of one there corresponds the product of the corresponding numbers of the other, the two systems are said to be abstractly equivalent or (simply) When two number systems are isomorphic, if any series of oper- ations is performed on numbers of one system and the same series of operations is performed on the corresponding numbers of the other, the resulting numbers will correspond. 53. Noiihomogeneous coordinates. By comparing the corollaries of Theorem 1 with the definition of group (p. 66), it is at once seen that the set of points of a line on which a scale has been estab- lished, forms a group with respect to addition, provided the point jg be excluded from the set. In this group J?J is the identity element, and the existence of an inverse for every element follows from Theorem 8. In the same way it is seen that the set of points on a line on which a scale has been established, and from which the * For the general idea of the isomorphism between groups, see Burnside's Theory of Groups, p. 22. 53] COOEDIXATES 151 point .g has been excluded, forms a group with respect to multipli- cation, except that no inverse with respect to multiplication exists for j?J; P^ is the identity element in this group, and Theorem 9 insures the existence of an inverse for every point except J. These con- siderations show that the first two conditions in the definition of a number system are satisfied by the points of a line, if the operations 9 and o are identified with addition and multiplication as defined in. 48 and 49. The third condition in the definition of a number system is also satisfied in view of Theorem 5. Finally, in view of Theorem 1, Cor. 4, and Theorem 6, this number system of points on a line is commutative with respect to both addition and multipli- cation. This gives then: THEOREM 10. TJie set of all points on o, line on which a scale has Tjeeii established, and from -ich ieli the point J2 is excluded, forms a field with res'pect to the operations of addition and multiplication preciously defined. (A, E, P) This provides a new way of regarding a point, viz.,, that of regarding a point as a nwnber of a number system. This conception of a point will apply to any point of a line except the one chosen as J. It is desirable, however, both on account of the presence of such an excep- tional point and also for other reasons, to keep the notion of point distinct from the notion of number, at least nominally. This we do by introducing a field of numbers a, I, c,- ,l,k,- , %,?/,%, which is isomorphie witli the field of points on a line. The numbers of the number field may, as we have seen, be the points of the line, or they may be mere symbols which combine according to the conditions specified in the definition of a number system ; or they may be ele- ments defined in some way in terms of points, lines, etc.* In any number system the identity element with respect to addi- tion is called zero and denoted by 0, and the identity element with respect to multiplication is called one or unity, and is denoted by 1. We shall, moreover, denote the numbers 1+1, 1+1+1, - - -, a, - by the usual symbols 2, 3, - - -, a, -.f In the isomorphism of our system of numbers with the set of points on a line, the point JJ must correspond to 0, the point JJ to the number 1 ; and, in general, to every * See, for example, 55, on von Staudt's algebra of throws, where the numbers are thought of as sets of four points. t Cf., however, in this connection 57 below. 152 ALQEBEA OF PODTTS [CHAP. VI point will correspond a number (except to j), and to every number of the field will correspond a point. In this way every point of the line (except .) is labeled by a number. This number is called the (nonhomogeneous) coordinate of the point, to which it corresponds. This enables us to express relations between points by means of equations between their coordinates. The coordinates of points, or the points themselves when we think of them as numbers of a number system, we will denote by the small letters of the alphabet (or by numerals), and we shall frequently use the phrase "the point a?" in place of the longer phrase " the point whose coordinate is x" It should be noted that this representation of the points of a line by numbers of a number system -is not in any way dependent on the commutativity of multiplication; i.e. it holds in the general geom- etries for which Assumption P is not assumed. Before leaving the present discussion it seems desirable to point out that the algebra of points on a line is merely representative, under the principle of duality, of the algebra of the elements of any one-dimensional primitive form. Thus three lines Z , l v ! of a flat pencil determine a scale in the pencil of lines; and three planes # , a v a:,, of an axial pencil determine a scale in this pencil of planes ; to each corresponds the same algebra. 54. The analytic expression for a projectivity in a one-dimensional primitive form. Let a scale be established on a line I by choosing three arbitrary points for P Q9 J%, , ; and let the resulting field of points on a line be made isomorphic with a field of numbers 0, 1, a, - , so that JJ corresponds to 0, ^ to 1, and, in general, P a to a. For the exceptional point -g, let us introduce a special symbol co with excep- tional properties, which will be assigned to it as the need arises. It should be noted here, however, that this new symbol oo does not represent a number of a field as defined on p. 149. We may now derive the analytic relation between the coordinates of the points on I, which expresses a projective correspondence between these points. Let x be the coordinate of any point of I. We have seen that if the point whose coordinate is x is made to correspond to either of the points (I) 3/=v + a, (a 3= oo) or (II) #'=a#, (a = 0) 54] LINEAR FRACTIONAL TRANSFORMATION 153 where a is the coordinate of any given point on /, each of the result- ing correspondences is projective (Theorem 2 and Theorem 4). It is readily seen, moreover, that if x is made to correspond to (in) *'=*, &,* the resulting correspondence is likewise projective. For we clearly have the following construction for the point l/,c (fig. 77): With the same notation as before for the construction of the product of two FIG. 77 numbers, let the line xA meet 7. in X. If T is determined as the intersection of IX with 7 , the line BY determines on Z a point x 1 , such that text =1, by definition. "We now have The three projectivities (I), (II), and (III) are of fundamental importance, as the next theorem will show. It is therefore desirable to consider their properties briefly ; we will thus be led to define the behavior of the exceptional symbol oo with respect to the operations of addition, subtraction, multiplication, and division. The projectivity a/= a? + a, from its definition, leaves the point %>, which we associated with co, invariant. We therefore place oo 4- a = oo for all values of a (a = oo). This projectivity, moreover, can have no other invariant point unless it leaves every point invariant ; for the equation x = x + a gives at once a = 0, if x 3= oo. Further, by prop- erly choosing a, any point x can be made to correspond to any point a/; 154 ALGEBKA OF POINTS [CHAP. YI but when one such pair of homologous points is assigned in addition to the double point cc, the projectivity is completely determined. The resultant or product of any two projectivities % f =x + a and #'= + & is clearly a/= x + (a -f &). Two such projectivities are therefore commutative. The pro jeetivity d = ax, from its definition, leaves the points and cc invariant, and by the fundamental theorem (Theorem 17, Chap. IV) cannot leave any other point invariant without reducing to the iden- tical projectivity. As another property of the symbol co we have therefore cc = a cc (a =r= 0). Here, also, by properly choosing a, any point x can be made to correspond to any point a/, biit then the pro- jectivity is completely determined. The fundamental theorem in this case shows, moreover, that any projectivity with the double points 0, co can be represented by this equation. The product of two projectivities x r = a:c and xf = 6.r is clearly ic r = (at) x, so that any two projectivities of this type are also commutative (Theorem 6). Finally, the projectivity x f =l/x, by its definition, makes the point co correspond to and the point to co. "We are therefore led to assign to the symbol co the following further properties : 1/co = 0, and 1/0 = x. This projectivity leaves 1 and 1 (defined as 1) invariant. Moreover, it is an involution because the resultant of two applications of this projectivity is clearly the identity; ie. if the projectivity is denoted by TT, it satisfies the relation 7r 2 = 1. THEOREM 11. Any projectivity o/i a line is the product of projec- timties of the three types (I), (JJ), and (III), and may be expressed in the form n\ rf (1) ' Conversely, every equation of this form represents a projectimty, if ad-lc^ 0. (A, E, P) Proof. We will prove the latter part of the theorem first. If we suppose first that c 3= 0, we may write the equation of the given transformation in the form , __ ad (2) ^.2 + 4. c cx + d This shows first that the determinant ad "be must be different from ; otherwise the second term on the right of (2) would vanish, which 54] LIXEAK FEACTIOXAL TEAXSFORMATIOX 155 would make every x correspond to the same point tt/c, while a pro- jectivity is a one-to-one correspondence. Equation (1), moreover, shows at once that the correspondence established by it is the result- ant of the five : ,7 1 /7 ''"^V / " x l ex, x n = #! + it, x s = > x = ( o -- .f.j, d x -T --- <>. 2 \ c f c If c = 0, and ad =?= 0, this argument is readily modified to show that the transformation of the theorem is the resultant of projectivities of the types (I) and (II). Since the resultant of any series of projectiv- ities is a projectivity, this proves the last part of the theorem. It remains to show that every projectivity can indeed be repre- sented bv an equation jsf = - - - To do this simplv, it is desirable - " ex + a to determine first what point is made to correspond to the point oc by this projectivity. If we follow the course of this point through the five projectivities into which we have just resolved this transforma- tion, it is seen that the first two leave it invariant, the third trans- forms it into 0, the fourth leaves invariant, and the fifth transforms it into a/c] the point oo is then transformed by (1) into the point a/c. This leads us to attribute a further property to the symbol cc, viz., , 7 a , - , when x = cc. c According to the fundamental theorem (Theorem 17, Chap. IV), a pro- jectivity is completely determined when any three pairs of homolo- gous points are assigned. Suppose that in a given projectivity the points 0, 1, oo are transformed into the points p, q> r respectively. Then the transformation 2) clearly transforms into p, 1 into gr, and, by virtue of the relation just developed for oo, it also transforms oo into r. It is, moreover, of the form of (1). The determinant adlc is in this case (j p)(r q}(r~p)> which is clearly different from zero, if p, q, r are all distinct. This transformation is therefore the given projectivity. COEOLLAEY 1. The projectivity sJ = a/x(a, = 0, or oo) transforms "into oo and oo into 0. (A, E, P) 156 ALGEBKA OF POINTS [CH^.TI For it is the resultant of the two projectivities, x l = l/x and x f = ax v of which the first interchanges and oo, while the second leaves them both invariant. We are therefore led to define the symbols a/Q and a/cc as equal to co and respectively, when a is neither no r oo. COROLLARY 2. Any projectivity leaving the point co invariant may 1e expressed in the form x f = ax + L (A, E, P) COROLLARY 3. A/iy projectivity may le expressed analytically "by the lilinear equation c%3/+ dx f ax~ 6 = 0; and conversely, any ~bilinec.tr equation defines a protective correspondence "between its two variables. (A, E, P) COROLLARY 4. If a projectixity leaves any points invariant, the coordinates of these doulle points satisfy the quadratic equation car + (da)x-1> = Q. (A, E, P) DEFINITION. A system of mn numbers arranged in a rectangular array of w rows and n columns is called a matrix. If m = n, it is called a square matrix of order n* The coefficients \ , ) of the protective transformation (1) form a \c aj * J v ' square matrix of the second order, which may be conveniently used to denote the transformation. Two matrices ( a , j and ( a f ) repre- sent the same transformation, if and only if a : a f == b : &'= c : e r = d : d f . The product of two projectivities , , . ax + 1 , , , v a f x f + V ' and z" =*7r.(x f )= . . 7 . is given by the equation x _ (aa 1 + cV) x + la 1 + db f - This leads at once to the rule for the multiplication of matrices, which is similar to that for determinants. DEFINITION. The product of two matrices is defined by the equation (a 1 V\ (a l\ _ (aa! + cV la} + db f \ \c f d f )\c d) \ac f +cd f I * For a development of the principal properties of matrices, cf . B6cher, Intro- duction to Higher Algebra, pp. 20 ff. 54,53] THROWS 157 This gives, in connection with the result just derived, THEOREM 12. The product of two projectmties is represented ly the product of their matrices; in sytnloh, COROLLAEY 1. The determinant of the 'product of tv:o 2)rojectivitie$ is equal to the product of their determinants. (A, E,.P) COROLLARY 2. The inverse of the projectimty 7r~i a \ is given d ~ l where A, B, C, D are the cofactors , of a, I, c, d respectively in- the determinant , . (A, E, P) This follows at once from Corollary 3 of the last theorem by inter- changing x 9 x 1 . We may also verify the relation by forming the product flrV= [ T 7 * V which transformation is equiva- \ aa oc/ lent to (~: !j ]. The latter is called the identical matrix. / 7% \ COROLLARY 3. Any involution is represented ly ( a ), that is \c -"" a I ly x f = 2i , with the condition that a?+t>c^ 0. (A, E, P) J ex a 55. Von Staudt's algebra of throws. "We wDl now consider the number system of points on a line from a slightly different point of view. On p. 60 we defined a throw as consisting of two ordered pairs of points on a line ; and defined two throws as equal when they are protective. The class of all throws which are protective (ie. equal) to a given throw constitutes a class which we shall call a mark. Every throw determines one and only one mark, but each mark determines a whole class of throws. According to the fundamental theorem (Theorem 17, Chap. IV), if three elements A 9 B, C of a throw and their pkces in the symbol T(A3, CD) are given, the throw is completely determined by the mark to which it belongs. A given mark can be denoted by the symbol of any onfe of the (projective) throws which define it. We shall also denote marks by the small letters of the alphabet. And so, since the equa]ity sign (=) indicates that the two symbols between 158 ALGEBEA OF POINTS [CHAP.YI which it stands denote the same thing, we may write T (AB, CD) = a = l t if a, I, T(AB, CD) are notations for the same mark. Thus T (AB, CD) = T (BA 9 DC) = T (CD, AB) = T(DC, BA) are all symbols denoting the same mark (Theorem 2, Chap. III). According to the original definition of a throw the four elements which compose it must be distinct. The term is now to be extended to include the following sets of two ordered pairs, where A, B, C are distinct. The set of all throws of the type 7(AB, CA) is called a mark and denoted by <x> ; the set of all throws of the type T(AB, CB) is called a mark and is denoted by ; the set of all throws of the type T(AB, CC) is a mark and is denoted by 1. It is readily seen that if %, %> j are any three points of a line, there exists for every point P of the line a unique throw T (_? JJ, JJ, P) of the line ; and con- versely, for every mark there is a unique point P. The mark co, by what precedes, corresponds to the point R> ; the mark to JJ ; and the mark 1 to J?. DEFINITION. Let T(AB, CDJ be a throw of the mark a, and let T(AB, <7D 2 ) be a throw of the mark &; then, if Z> 3 is determined by iB, ADJ>^ 9 the mark c of the throw T(AB 9 CD S ) is called the of the marks a and &, and is denoted by a + b; in symbols, a -f- J = c. Also, the point D[ determined by Q (AD^C y BDJD^ deter- mines a mark with the symbol T(AB, CD f B ) = c r (say), which is called the product of the marks a> and 6 ; in symbols, ab = c 7 . As to the marks and 1, to which these two definitions do not apply, we define further : #-f-0 = + # = , ^-0 = 0-^ = 0, and a - 1 = 1 a = a. Since any three distinct points A } B, C may be projected into a fixed triple j, ^ JJ, it follows that the operation of adding or multiplying marks may be performed on their representative throws of the form T(%>PQ> JiP)- By reference to Theorems 1 and 3 it is then clear that the class of all marks on a line (except oo) forms a number system, with respect to the operations of addition and multiplication just defined, which is isomorphic with the number system of points previously developed. This is, in brief, the method used by von Staudt to introduce ana- lytic methods into geometry on a purely geometric basis.* "We have * Cf . reference on p. 141. Von Staudt used the notion of an involution on a line in defining addition and multiplication ; the definition in terms of quadrangular sets is, however, essentially the same as his by virtue of Theorem 27, Chap. IV. 5o,oG] CROSS RATIO 159 given it here partly on account of its historical importance; partly because it gives a concrete example of a number system isoinurpliic with the points of a line * ; and partly because it gives a natural introduction to the fundamental concept of the cross ratio uf four points. This we proceed to derive in the next section. 56. Ttie cross ratio. We have seen in the preceding section that it is possible to associate a number with every throw of four points on a line. By duality all the developments of this section apply also to the other one-dimensional primitive forms, i.e. the pencil of lines and the pencil of planes. TTith every throw of four elements of any one-dimensional primitive form there may be associated a definite number, which must be the same for every throw protective with the first, and is therefore an invariant under any protective transforma- tion, i.e. a property of the throw that is not changed when the throw is replaced by any protective throw. This number is called the cross ratio of the throw. It is also called the doulle ratio or the anhar- monic ratio. The reason for these names will appear presently. In general, four given points give rise to six different cross ratios. For the 24 possible permutations of the letters in the symbol T(AB, CD} fall into sets of four which, by virtue of Theorem 2, Chap. Ill, have the same cross ratios. In the array below, the per- mutations in any line are protective with each other, two permuta- tions of different lines being in general not projective : AB, CD BA, DC DC, BA CD, AB AB, DC BA, CD CD, BA DC, AB AC, BD CA, DB DB, CA BD, AC AC, DB CA, BD BD, CA DB, AC AD, BC DA, CB .CB, DA BC, AD AD, CB DA, BC BC, DA CB, AD If, however, the four points form a harmonic set H (AB, CD), the throws T(AB, CD) and T(AB, DC) are projective (Theorem 5, Cor. 2, Chap. IV). In this case the permutations in the first two rows of the array just given are all projective and hence have the same cross ratio. The four elements of a harmonic set, therefore, give rise to only three crostTratios. The values of these cross ratios are readily seen * Cf. 53. Here, with every point of a line on which a scale has been estab- lished, is associated a mark which Js the coordinate of the point. 160 ALQEBEA OF POINTS [CHAP.TI to be 1, J-, 2 respectively, for the constructions of our number system give* at once H (gJJ, JJP 1 ) 9 H (JJ, 1J1J), and H ( jfj, JJJ>). We now proceed to develop an analytic expression for the cross ratio "Be, (# r r a , # 3 : 4 ) of any four points on a line (or, in general, of any four elements of any one-dimensional primitive form) whose coordi- nates in a given scale are given. It seems desirable to precede this derivation by an explicit definition of this cross ratio, which is inde- pendent of von Staudt's algebra of throws. DEFINITION. The cross ratio I&.fax^ x 9 x^ of elements x v # a , # 8 , x of any one-dimensional form is, if x 19 # a , x z are distinct, the coordi- nate X of the element of the form into which # 4 is transformed by the projectivity which transforms x v x s , x s into oo, 0, 1 respectively ; i.e. the number, X, defined by the projectivity o^oy^-^coOlX. If two of the elements x v # 2 , # 8 coincide and x is distinct from all of them, we define E (x^, x 3 x) as that one of B (x 2 x 19 x^), B (# 8 # 4 , jTjA'j,), Ii (j: 4 r 8 , ayj, for which the first three elements are distinct. THEOREM 13. TJie cross ratio B (x^x^ x z x^ of the four dements whose coordinates are respectively x v % 2 , x s> x is given ~by the relation \ I "~~" - (A, E, P) Proof, The transformation is evidently a projectivity, since it is reducible to the form of a linear fractional transformation, viz., in which the determinant (a^ c s ) (# 2 ^ S )(cc 3 ^ is not zero, pro- vided the points x v % Q> x s are distinct. This projectivity transforms x v x 2 , X B into oo, 0, 1 respectively. By definition, therefore, this pro- jectivity transforms x into the point whose coordinate is the cross ratio in question, Le. into the expression given in the theorem. If x v x 2 , X B are not all distinct, replace the symbol 1$ (ayj a , x z x^) by one of its equal cross ratios B? (x z x v x^), etc. ; one of these must have the first three elements of the symbol distinct, since in a cross ratio of four points at least three must be distinct (def.). 56] OEOSS BATIO 161 COROLLARY 1. We have in particular B Uy>: 2 , ,0,'j) = cc, B (.<.y\, V*.' 2 ) ^ an $ ^ ( i W W = 1 -z/^, c 2 , # 3 are fl/iy rtra? distitid dements of the form. (A, E) COROLLARY 2. T&e ?*0s$ /Y^/O ^/ hanno/iic set H(ay: 2 , a.y 4 ) Z6 5 B (^yj 2 , iy; 4 ) == 1, for we have H (ac 0, 1 1 ). (A, E s P) "COROLLARY 3. J/ R (x*^ ^v^) = X, rt^ other five cross ratios of the throws composed of the fovr dements x v # 2 , .r s , A' 4 /? 5 (' Vs ;<: A) = ' E ( V ! * A v a) ^ ~- ' ,,, 4fl (A, E, P) The proof is left as an exercise. COROLLARY 4. If x v # 2 , C 3 , x form a harmonic set H (j/^, (A, E, P) The proof is left as an exercise. COROLLARY 5. If a, I, c are any three distinct elements of a one- dimensional primitive form, and a 1 , b', d are any three other distinct elements of the same form, then the correspondence established ~by the relation E (ab 9 ex) = B (aV, c ! x r ) is projective. (A, E, P) Proof. Analytically this relation gives a c b x__a r c f Vx* a x 1) c a f x f U c f which, when expanded, evidently leads to a bilinear equation in the variables x, x f , which defines a projective correspondence by Theorem 11, Cor. 3. That the cross ratio is invariant under any projective transformation may also be verified directly by observing that each of the three types (I), (II), (III) of projectivities on pp. 152, 153 leaves it invariant. That every projectivity leaves it invariant then follows from Theorem 11. 162 ALGEBRA OF POINTS [CHAP.YI 57. Coordinates in a net of rationality on a line. We now con- sider the numbers associated with the points of a net of rationality on a line. The connection between the developments of this chapter and the notion of a linear net of rationality is contained in the following theorem : THEOREM 14. The coordinates of the points of the net of rationality R(JJJ.B)/0m a number system, or field, which consists of all numbers each of tchich can be obtained by a finite number of rational algebraic operations o/i and 1, and only these. (A, E) Proof. By Theorem 14, Chap. IV, the linear net is a line of the rational space constituted by the points of a three-dimensional net of rationality. By Theorem 20, Chap. IV, this three-dimensional net is a properly protective space. Hence, by Theorem 10 of the present chapter, the numbers associated with R(Oloo) form a field. All numbers obtainable from and 1 by the operations of addi- tion, subtraction, multiplication, and division are in R(Olco), because (Theorem 9, Chap. IV) whenever x and y are in R (Oloo) the quadran- gular sets determining x + y, xy, % y, x/y have five out of six elements in R(Oloc). On the other hand, every number of R(Oloo) can be obtained by a finite number of these operations. This follows from the fact that the harmonic conjugate of any point a in R(Oloo) with respect to two others, 5, c, can be obtained by a finite number of rational operations on a, I, c. This fact is a consequence of Theo- rem 13, Cor. 2, which shows that x is connected with a, b, c by the relation (x b) (a c) -f (x c) (a b) = 0. Solving this equation for x, we have 2 be ab ac a number * which is clearly the result of a finite number of rational operations on a, 1 9 c. This completes the proof of the theorem. We have here the reason for the term net of rationality. It is well to recall at this point that our assumptions are not yet sufficient to identify the numbers associated with a net of rationality with the system of all ordinary rational numbers. We need only recall the example of the miniature geometry described in the Introduction, 2, which contained only * The expression for x cannot be indeterminate unless & = c. 57, os] HOMOGENEOUS COORDINATES 163 three points on a line. If in that triple-system geometry we perform the con- struction for the number 1 + 1 on any line in which we have assigned the numbers 0, 1, to the three points of the Hue in any way, it will be found that this construction yields the point 0. This, is due to the fact previously noted that in that geometry the diagonal points of a complete quadrangle are collinear. In every geometry to which Assumptions A, E, P apply we may construct the points 1 + 1, 1 + 1 + 1 thus forming a sequence of points which, with the usual notation for these sums, we may denote by 0, 1, 2, 3, 4, .--. Two possibilities then present themselves: either the points, thus obtained are all distinct, in ^liich case the net R(Olcc) contains all the ordinary rational numbers ; or some point of this sequence coincides with one of the preceding points of the sequence, in which case the number of points in a net of rationality is finite. We shall consider this situation in detail in a later chapter, and will then add further assumptions. Here it should be emphasized that our results hitherto, and all subsequent results depending only on Assumptions A, E, P, are valid not only in the ordinary real or complex geometries, but in a much more general class of spaces, which are character- ized merely by the fact that the coordinates of the points on a line are the numbers of a field, finite or infinite. 58. Homogeneous coordinates on a line. The exceptional character of the point J2, as the coordinate of which we introduced a symbol co with exceptional properties, often proves troublesome, and is, more- over, contrary to the spirit of protective geometry in which the points of a line are all equivalent ; indeed, the choice of the point & was entirely arbitrary. It is exceptional only in its relation to the opera- tions of addition, multiplication, etc., which we have defined in terms of it. In this section we will describe another method of denoting points on a line by numbers, whereby it is not necessary to use any exceptional symbol. As before, let a scale be established on a line by choosing any three points to be the points P^ J?, R ; and let each point of the line be denoted by its (nonhomogeneous) coordinate in a number system isomorphic with the points of the line. We will now associate with every point a pair of numbers (x v # 2 ) of this system in a given order, such that if x is the (nonhomogeneous) coordinate of any point dis- tinct from Jg,, the pair (x v x^f associated with the point is satisfies the relation x = xj%y. TVith the point R we associate any pair of the form (&, 0), where k is any number (&=0) of the number system isomorphic with the line. To every jwint of ilic line corresponds a pair of numbers, and to every pair of numbers in the field, except fJie pair 164 ALGEBPtA OF POINTS [CHAP.TI (0, 0), corresponds a unique poiiit of the line. These two numbers are called homogeneous coordinates of the point with which they are associated, and the pair of numbers is said to represent the point. This representation of points on a line by pairs of numbers is not unique, since only the ratio of the two coordinates is determined; Le. the pairs (x r r 2 ) and (mx v mx 2 ) represent the same point for all values of m different from 0. The point P Q is characterized by the fact that ^ = ; the point H by the fact that # 2 = ; and the point % by the fact that x\ x y THEOREM 15. Li homogeneous coordinates a projectwity on a line is represented by a linear homogeneous transformation in two variables, (1) p*l~**i + lp, (orf-fc^O) V pZJ = 0^+. d%, where p is an arbitrary factor of proportionality. (A, E, P) Proof. By division, this clearly leads to the transformation cx + d provided x^ and # 2 are both different from 0. If # 3 = 0, the trans- formation (1) gives the point (a?/, a?/) = (a t c) ; i.e. the point H = (1, 0) is transformed by (1) into the point whose nonhomogeneous coordinate is a/c. And if # 2 '=0, we have in (1) (x p % 2 ) = (d, c); i.e. (1) transforms the point whose nonhomogeneous coordinate is rf/e into the point R. By reference to Theorem 11 the validity of the theorem is therefore established. As before, the matrix ( a , ) of the coefficients may conveniently be used to represent the projectivity. The double points of the pro- jectivity, if existent, are obtained in homogeneous coordinates as follows : The coordinates of a double point (& v # 2 ) must satisfy the equations px ^ = aXl + bz z , px z =cx : +dXy These equations are compatible only if the determinant of the system vanishes. This leads to the equation c d *~~ p 58] HOMOGENEOUS COORDINATES 165 for the determination of the factor of proportionality p. This equa- tion is called the characteristic equation of the matrix representing the projeetivity. Every value of p satisfying this equation then leads to a double point when substituted in one of the equations (3) ; viz., if p 1 be a solution of the characteristic equation, the point ( t /: r :c z ) = (- 5, a - pj = (d - p v - c) is a double point."* In homogeneous coordinates the cross ratio TkiAB, CD) of four points A = (a v 2 ), B = (l v & 2 ), C = (c v <g, D = (d v rf 2 ) is given by (ad) (bd) where the expressions (ac) 9 etc., are used as abbreviations for Q-fia l9 etc. This statement is readily verified by writing down the above ratio in terms of the nonhomogeneous coordinates of the four points. We will close this section by giving to the two homogeneous coor- dinates of a point on a line an explicit geometrical significance. In view of the fact that the coordinates of such a point are not uniquely determined, a factor of proportionality being entirely arbitrary, there may be many such interpretations. On account of the existence of this arbitrary factor, we may impose a further condition on the coor- dinates (x v # 2 ) of a point, in addition to the defining relation xjx^x, where x is the nonhomogeneous coordinate of the point in question, We choose the relation x^+ # 2 = 1. If this relation is satisfied, = B( 10, oo a?), = B( loo, 00). Thus homogeneous coordinates subject to the condition o^ 4- # 2 = 1 can be defined by choosing three points A 9 B, C arbitrarily, and letting x l = & (AB, CX} and a? 2 = B (AC, BX). The ordinary homogeneous coordinates would then be defined as any two numbers proportional to these two cross ratios. * This point is indeterminate only if 5 = c = and a = d. The projectiyity is then the identity. 1 -1 1 1 1 1 -1 1 #! X 2 ! SS 2 I -I I 1 1 1 -1 1 166 ALGEBRA OE POINTS [CHAP. VI 59. Projective correspondence between the points of two different lines. Hitherto we have confined ourselves, in the development of analytic methods, to the points of a single line, or, under duality, to the elements of a single one-dimensional primitive form. Suppose now that we have two lines I and m with a scale on each, and let the nonhomogeneous coordinate of any point of I be represented by x, and that of any point of m by y. The question then arises as to how a projective correspondence between the point x and the point y may be expressed analytically. It is necessary, first of all, to give a meaning to the equation y = x. In other words : What is meant by say- ing that two points $ on /, and y on m have the same coordinate ? The coordinate a? is a number of a field and corresponds to the point of which it is the coordinate in an isomorphism of this field with the field of points on the line 1. "We may think of this same field of numbers as isomorphic with the field of points on the line m. In bringing about this isomorphism nothing has been specified except that the fundamental points J, Q . determining the scale on m must correspond to the numbers 0, 1 and the symbol oo respectively. If the correspondence between the points of the line and the numbers of the field were entirely determined by the respective correspond- ences of the points ^, JJ, R> just mentioned, then we should know precisely what points on the two lines I and m have the same coor- dinates. It is not true of all fields, however, that this correspondence is uniquely determined when the points corresponding to 0, 1, oo are assigned.* It is necessary, therefore, to specify more definitely how the isomorphism between the points of m and the numbers of the field is brought about. One way to bring it about is to make use of the projectivity which carries the fundamental points 0, 1, oo of I into the fundamental points 0, 1, cc of m, and to assign the coordinate & of any point A of I to that point of m into which A is transformed by this projectivity. In this projectivity pairs of homologous points will then have the same coordinates. That the field of points and the field of numbers are indeed made isomorphic by this process follows directly from Theorems 1 and 3 in connection with Theorem 1, Cor., Chap. IV. TFe may now readily prove the following theorem : * This is shown by the fact that the field of all ordinary complex numbers can be isomorphic with itself not only by making each number correspond to itself, but also by making each number a + ib> correspond to its conjugate a i&. 59] EXERCISES 167 THEOREM 16. Any projecthe correspondence between the points [:e] and [y] of two distinct lines may le represented analytically by tlu relation y x 'by properly choosing the coordinates on the two lines. If the coordinates on the two lines are so related that tltc relation y = x represents a projectile correspondence, then any projective cor- respondence between the points of the tico lines is given by a relation CM + T> ,11 y = _ , f cu ( fa -fc Q\ C& + d (A, E, P) Proof. The first part of the theorem follows at once from the pre- ceding discussion, since any projectivity is determined by three pairs of homologous points, and any three points of either line may be chosen for the fundamental points. In fact, we may represent any projectivity between the points of the two lines by the relation y = x, by choosing the fundamental points on I arbitrarily; the fundamental points on m are then uniquely determined. To prove the second part of the theorem, let TT be any given projective transformation of the points of the line I into those of m, and let TT O be the projectivity y sc, regarded as a transformation from m to /. The resultant TTJIT = ^ is a projectivity on ?, and may therefore be represented by se/ = (ax + 1) /(ex + d). Since TT = ir~ V a , this gives readily the result that TT may be represented by the relation given in the theorem. EXERCISES 1. Give constructions for subtraction and division in the algebra of points on a line. 2. Give constructions for the sum and the product of two lines of a pencil of lines in which a scale has been established. 3. Develop the point algebra on a line by using the properties expressed in Theorems 2 and 4 as the definitions of addition and multiplication respec- tively. Is it necessary to use Assumption P from the beginning? 4. Using Cor. 3 of Theorem 9, Chap. Ill, show that addition and multi- plication may be defined as follows : As before, choose three points P , P v P on a line Z as fundamental points, and let any line through Poo be labeled 2,0. Then the sum of two numbers P x and P y is the point P x+ y into which P v is transformed by the elation with axis I* and center P m which transforms P into P.,.; and the product P^-Pj, is the point P^ into which P y is trans- formed by the homology with axis Z w and center P which transforms P l into P.,,. Develop the point algebra on this basis without using Assumption P, except in the proof of the commutativity of multiplication. 168 ALGEBRA OF POIXTS [CHAP.TI 5. If the relation ax = ly holds between four points a, 5, x, y of a line, show that we have Q(0fca, X#JT). Is Assumption P necessary for this result? 6. Prove l>v direct computation that the expression -^ ^ : -8 % i s fc *>* 7* 7* - ** *1 -^ *g ^4 unchanged in value when the four points x v x 2 , # s? o? 4 are subjected to any- linear fractional transformation u/ = cx + d 7. Prove that the transformations X' = X, X' = ~, X' = 1-X, X'=-i-r, *' = r^7' *' = ^ A I A A "" JL A form a group. What are the periods of the various transformations of this group? (Cf. Theorem 13, Cor. 3.) 8. If A, B, C, P v P 2 , , P n are any n -f 3 points of a line, show that every cross ratio of any four of these points can be expressed rationally in terms of the n cross ratios A, = B (AB< CP ), i = 1, 2, , n. When n = 1 this reduces to Theorem 13, Cor. 3. Discuss in detail the case n = 2. 9. If B: (a^o, x^x 4 ) = X, show that 1-X = 1 X ^-^"^-^2 *3-*l' The relation of Cor. 3 of Theorem 13 is a special case of this relation. 10. Show that if R (AB, CD) = E (AB< DC), the points form a harmonic set H (-43, CD). 11. If the cross ratio E (.15, CD) = X satisfies the equation X 2 -X + 1 = 0, then B (.IB, CD) = B (4 C 5 D) = B (AD, JJC) = X, and B (AB.DC) = B (-4 C, BD) = B (4D, CB) = - X 2 . 12. If A, B, T, F, Z are any five points on a line, show that B (-4B, AT) B (4B, rZ) B (-4B, ZZ) = 1. 13. State the corollaries of Theorem 11 in homogeneous coordinates. 14. By direct computation show that the two methods of determining the double points of a projectivity described in 54 and 58 are equivalent. 15. If Q(lBC,ATZ),then B (AX, FC) + B (BY, ZA) -f B (CZ 9 ZB) = 1. 16. If M v Jf 2 , J/ 3 are any three points in the plane of a line I but not on L the cross ratios of the lines ?, -PJ/p PM^ PM Z are different for any two points P on L 17. If A , B are any two fixed points on a line Z, and X, F are two variable points such that B (AB, XY) is constant, the set [Z j is projective with the set [F]. CHAPTER VII COORDINATE SYSTEMS IN TWO- AND THREE-DIMENSIONAL* FORMS 60. Nonhomogeneotts coordinates in a plane. In order to repre- sent the points and lines of a plane analytically we proceed as follows : Choose any two distinct lines of the plane, which we will call the axes of coordinates, and determine on each a scale (48) arbitrarily, except that the point of intersection of the lines shall be the 0-point on each scale (fig. 78). This point we call the origin. Denote the fundamental points on one of the lines, which we call the x-axis } by O a , I,,., oOp ; and on the other line, which we will call the y-axis, by O y , l y , oo y . Let the line cc^cc, be de- ^ Tg noted by Z. Now let P be any point in the plane not on Z. Let the lines Pcc y and Poo,,, meet the #-axis and the y-axis in points whose nonhornoge- neous coordinates are a and ~b respectively, in the scales just estab- lished. The two numbers a, "b uniquely determine and are uniquely determined by the point P. Thus every point in the plane not on l m is represented by a pair of numbers ; and, conversely, every pair of numbers of which one belongs to the scale on the as-axis and the other to the scale on the y-axis determines a point in the plane (the pair of symbols co^ oo y being excluded). The exceptional character of the points on l m mil be removed presently ( 63) by considera- tions similar to those used to remove the exceptional character of *A11 the developments of this chapter are on the basis of Assumptions A,E,P. 169 170 COORDINATE SYSTEMS [CHAP, vn the point cc in the case of the analytic treatment of the points of a line ( 58). The two numbers just described, determining the point P, are called the nonhomogencous coordinates of P with reference to FIG. 79 the two scales on the x- and the ^-axes. The point P is then repre- sented analytically by the symbol ( , 6). The number a is called the ^-coordinate or the abscissa of the point, and is always written first in the symbol representing the point; the number & is called the ^/-coordinate or the ordinate of the point, and is always written last in this symbol. The plane dual of the process just described leads to the corre- sponding analytic representation of a line in the plane. For this pur- pose, choose any two distinct points in the plane, which we will call the centers of coordinates ; and in each of the pencils of lines with these centers determine a scale arbitrarily, except that the line o join- ing the two points shall be the 0-line in each scale. This line we call the origin. Denote the fundamental lines on one of the points, which we will call the u-center, by O tt , l a , o> tt ; and on the other point, which we will call the v-center, by O p , l r , cc p . Let the point of intersection of the lines oo w , oo r be denoted by & (fig. 79). Sow let I be any line in the plane not on R. Let the points lco v and /oc tt be on the lines of the ^-center and the ^-center, whose non- homogeneous coordinates are m and n respectively in the scales just established. The two numbers m, n uniquely determine and are uniquely determined by the line L Thus every line in the plane not on is represented by a pair of numbers ; and, conversely, every pair of numbers of which one belongs to the scale on the ^-center and the other to the scale on the ^-center determines a line in the plane (the pair of symbols OO M , oo p being excluded). The exceptional character 60, 01] COORDINATES IS A PLANE 171 of the lines on j will also be removed presently. The two numbers just described, determining the line /, are called the nonhomogcueous coordinates of I with reference to the two scales on the it- and ^-centers. The line I is then represented analytically by the symbol \m, %]. The number m is called the u-co'drdinate of the line, and is always written first in the symbol just given ; the number 11 is called the ^-coordinate of the line, and is always written second in this symbol. A variable point of the plane will frequently be represented by the symbol (x 9 y) ; a variable line by the symbol [u, r]. The coor- dinates of a point referred to two axes are called point coordinates ; the coordinates of a line referred to two centers are called line coor- dinates. The line Z M and the point R are called the singular line and the singular point respectively. 61* Simultaneous point and line coordinates. In developing further our analytic methods we must agree upon a convenient relation between the axes and centers of the point and line coordinates respec- tively. Let us consider any triangle in the plane, say with vertices 0, Z7, V. let the lines OU and OF be the y- and 0-axes respectively, and in establishing the scales on these axes let the points 7, V be the points cc y , co a respectively (fig. 80), "Further, let the points J7, 7 be the ^-center and the ^-center respectively, and in establishing the 172 COORDINATE SYSTEMS [CHAP, vn scales on these centers let the lines UO, TO be the lines co u , cc v respectively. The scales are now established except for the choice of the 1 points or lines in each scale. Let us choose arbitrarily a point I,, on the #-axis and a point l y on the y-axis (distinct, of course, from the points 0, U, F). The scales on the axes now being determined, we determine the scales on the centers as follows : Let the line on V and the point l x on the a-axis be the line l u ; and let the line on F and the point l v on the y-axis be the line l p . All the scales are now fixed. Let T be the projectivity (59, Chap. VI) between the points of the i'-axis and the lines of the w-center in which points and lines correspond when their x- and z^-coordinates respectively are the same. If TT' is the perspeetivity in which every line on the ^-center corresponds to the point in which it meets the #-axis, the product TT'TT transforms the <?>axis into itself and interchanges and GO.,, and I, and l x . Hence TrV is the involution a/ = 1/#. Hence it follows that the line on U whose coordinate is u is on the point of the x-axis whose coordinate is 1/u; and the point on the x-axis whose coordinate is x is oil the line of the u-center whose coordinate is I/a*. This is the relation between the scales on the #-axis and the 2^-center. Similar considerations with reference to the y-axis and the -^-center lead to the corresponding result in this case : The line on V 'whose coor- dinate is v is on that point of the y-axis whose coordinate is 1/v; and the point of the y-axis whose coordinate is y is on that line of the v-ceiiter whose coordinate is 1/y. 62. Condition that a point be on a line. Suppose that, referred to a system of point-and-line coordinates described above, a point P has coordinates (a, V) and a line / has coordinates \m, n]. The condition that P be on I is now readily obtainable. Let us suppose, first, that none of the coordinates a, I, m, n are zero. We may proceed in either one of two dual ways. Adopting one of these, we know from the results of the preceding section that the line [m, n] meets the #-axis in a point whose ^-coordinate is 1/m, and meets the 7/-axis in a point whose ^-coordinate is 1/n (fig. 81). Also, by definition, the line joining P = (a, 1) to V meets the #-axis in a point whose ^-coor- dinate is a ; and the line joining P to Z7 meets the y-axis in a point whose y-coorclinate is &. If P is on I, we clearly have the following perspeetivity : 62] COOEDINATES IN A PLANE 173 = A (1) Hence we have which, when expanded (Theorem 13, Chap. VI), gives for the desired condition (3) ma + nl + 1 = 0. This condition has been shown to be necessary. It is also sufficient, for, if it is satisfied, relation (2) must hold, and hence would follow (Theorem 13, Cor. 5, Chap. TI) -- Oacc x j- -- Oac A m A n v But since this projectivity has the self-corresponding element 0, it is a perspectivity which leads to relation (1). But this implies that P is on L FIG. 81 If now & = (&= 0), we have at once 5 = l/n ; and if & = (a^ 0), we have likewise a = 1/m for the condition that P be on But each of these relations is equivalent to (3) when a = and J = respectively. The combination a = 0, 5 = gives the origin which is never on a line [m, ?i] where m^O^n. It follows in the same way directly from the definition that relation (3) gives the desired condition^ if we have either m = or n = 0. The condition (3) is then valid for all cases, and we have 174 COOEDIXATE SYSTEMS [CHAP.VH THEOREM 1. The necessary and sufficient condition that a point P = (a, 6) le on a line I = [?n, n] is that the relation ma + ^5 + 1 = be satisfied. DEFINITION. The equation DEFINITION. The equation which is satisfied by the eoordi- which is satisfied by the coordi- nates of all the points on a given nates of all the lines on a given line and no others is called the point and no others is called the point equation of the line. line equation of the point. COEOLLARY 1. The point equa- COROLLAEY 1'. The line equa- tion of the line [m, n] is tion of the point (a, V) is m + ny + 1 = 0. au + bv + 1 = 0. EXERCISE Derive the condition of Theorem 1 by dualizing the proof given. 63. Homogeneous coordinates in the plane. In the analytic repre- sentation of points and lines developed in the preceding sections the points on the line Z7F=0 and the lines on the point were left unconsidered. To remove the exceptional character of these points and lines, we may recall that in the case of a similar problem in the analytic representation of the elements of a one-dimensional form we found it convenient to replace the nonhomogeneous coordinate x of a point on a line by a pair of numbers x v # 2 whose ratio xjx^ was equal to x(x = oo), and such that x z = when x = oo. A similar system of homogeneous coordinates can be established for the plane. Denote the vertices 0, U 9 V of any triangle, which we will call the triangle of reference, by the " coordinates " (0, 0, 1), (0, 1, 0), (1, 0, 0) respectively, and an arbitrary point T, not on a side of the triangle of reference, by (1, 1, 1). The complete quadrangle OUVT is called the frame of reference * of the system of coordinates to be established. The three lines UT, VT, OT meet the other sides of the triangle of reference in points which we denote by l x = (1, 0, 1), 1 F =(0, 1, 1), 1,='(1, 1, 0) respectively (fig. 82). We will now show how it is possible to denote every point in the plane by a set of coordinates (x v # 2 , # 8 ). Observe first that we have thus far determined three points on each of the sides of the triangle * Frame of reference is a general term that may be applied to the fundamental elements of any coordinate system. 63] HOMOGENEOUS COORDINATES 175 of reference, viz.: (0, 0, 1), (0, 1, 1), (0, 1, 0) on OU; (0, 0, 1), (1, 0, 1), (1, 0, 0) on OF; and (0, 1, 0), (1, 1, 0), (1, 0, 0) on CT. The coordi- nates which we have assigned to these points are all of the form (x v # 2 , # 3 ). The three points on OU are characterized by the fact that x l = 0. Fixing attention on the remaining coordinates, we choose the points (0, 0, 1), (0, 1, 1), (67 X 0} as the fundamental points (0, I)/ (1, 1), (1, 0) of a system of homogeneous coordinates on the line OU. If in this system a point has coordinates (/, 7/1), we denote it in our planar system by (0, 1, m). In like manner, to the points of the other two sides of the triangle of reference may be assigned coordinates of the form (Is, 0, m) and (k, I, 0) respectively. We have thus assigned coordinates of the form (x v x^ r s ) to aH the points of the sides of the triangle of reference. Moreover, the coordinates of every point on these sides satisfy one of the three relations 2^= 0, ^ 2 = 0, # 3 = 0. Now let P be any point in the plane not on a side of the triangle of reference. P is uniquely determined if the coordinates of its pro- jections from any two of the vertices of the triangle of reference on the opposite sides are known. Let its projections from U and V on the sides OY and OU be (k, 0, n) and (0, 1', n r ) respectively. Since under the hypothesis none of the numbers k> %, l f , n f is zero, it is clearly possible to choose three numbers (x v % 2 , x z ) such that x l : x 3 = &:?&, and # a : x z = V : n f . We may then denote P by the coordinates (x v x# x s ). To make this system of coordinates effective, however, we must show that the same set of three numbers (x v x & x & ) can be obtained by projecting P on any other pair of sides of the triangle of reference. In other words, we must show that the projection of P = (x^ x# x s ) from on the line TJY is the point (x v x 2 , 0). Since this is clearly true of the point jP = (l, 1, 1), we assume P distinct from 51 Since the numbers # 1} # 2 , % 3 are all different from 0, let us place x 1 : x z = a?, and # 2 : # 3 = y, so that x and y are the nonhomoge- neous coordinates of (x v 0, # 3 ) and (0, # 2 , x s ) respectively in the scales on 07 and OU defined by = O x , l a , F= oo r and = O y , l y , U= ay Finally, let OP meet U Y in the point whose nonhomogeneous coor- dinate in the scale defined by ?7= O z , l z , F= co z is %} and let OP meet the line 1 X U in A. We now have V * * * yf * z - J y^y y > 176 COOBDIKATE SYSTEMS [CHAP.VH where C is the point in which TA meets OU. This projectivity between the lines UV and OU transforms O s into co y , cc s into O y , and 1 3 into l y . It follows that C has the coordinate 1/2 in the scale on Or. We have also which gives = B Substituting x = ^ : # 3 , and ?/ = # 3 : <? 3 , this gives the desired relation z = x t : </.' follows : z = x t : </.' 2 . The results of this discussion may be summarized as FIG. 82 THEOREM 2. DEFINITION. If P is any point not on a side of the triangle of reference OUV, there exist three numbers x v x z > x s ( a ^ Dif- ferent from 0) such that the projections of P from the vertices 0.17. T on the opposite sides have coordinates (x v x^ 0), (x v 0, # 8 ), (0, # 2 , # 8 ) respectively. TJiese three numbers are called the homogeneous coordf- "nafes^qflP, and P is denoted ly (x v x^ x^ Any set of three numbers (not all equarto'Q) determine uniquely a point whose (homogeneous) coordinates they are. The truth of the last sentence in the above theorem follows from the fact that, if one of the coordinates is 0, they determine uniquely a point on one of the sides of the triangle of reference ; whereas, if none is equal to 0, the lines joining U to (x v 0, # 8 ) and V to (0, # 2 ,j*! 8 ) meet in a point whose coordinates by the reasoning above are (x v x y xj). 63] HOMOGENEOUS COOEDDsATES ITT COKOLLAKY. The coordinates (x v 2 2 , z s ) and (l\c v k% 2 , hc 2 ) tktcmiiiu the same point, ifk is not 0. Homogeneous line coordinates arise by dualizing the above discus- sion in the plane. Thus we choose any quadrilateral in the plane as frame of reference, denoting the sides by [1, 0, 0], [0, 1, 0], [0, 0, 1], [1, 1, 1] respectively. The points of intersection with [1, 1, 1] of the lines [1, 0, 0], [0, 1, 0], [0, 0, 1] are joined to the vertices of the tri- angle of reference opposite to [1, 0, 0], [0, 1, 0], [0, 0, 1] respectively by lines that are denoted by [0, 1, 1], [1, 0, 1], [1, 1, 0]. The three lines [1, 0, 0], [1, 1, 0], [0, 1, 0] are then taken as the fundamental lines [1, 0], [1, 1], [0, 1] of a homogeneous system of coordinates in a flat penciL If in this system a line is denoted by [u v wj, it is denoted in the planar system by [w 1? 2 , 0]. In like manner, to the lines on the other vertices are assigned coordinates of the forms [0, ^ 2 , J and [u v 0, J respectively. As the plane dual of the theorem and definition above we then have at once THEOREM 2 f . DEFINITION. If 1 is any line not on a, vertex of the triangle of reference, there exist three numbers u v u 09 u s all different from zero, such that the traces of I on the three sides of the triangle of reference are projected from the respective opposite, vertices by the lines \u v u z) 0], [u v 0, ttj, [0, 2 , ?/]. FJiese three numbers are called the homogeneous coordinates of /, and I is denoted ly [u 19 u^ w s ]. Any set of three numbers (not all zero) determine uniquely a line whose coordinates they are. Homogeneous point and line coordinates may be put into such a relation that the condition that a point (x^ % 9 or s ) be on a line [!, a , tt s ] is that the relation u^+ ^ 2 # 3 -f u 3 x s = be satisfied. We have seen that if (x v x v x 9 ) is a point not on a side of the triangle of reference, and we place x = xjx z> and y = # 2 /# 3 , the numbers (x, y) are the nonhomogeneous coordinates of the point (x v x s , a? 3 ) referred to OF as the re-axis and to OU as the y-asis of a system of nonho- mogeneous coordinates in which the point T=(l, 1, 1) is the point (1, 1) (0, U, 7 being used in the same significance as in the proof of Theorem 2). By duality, if [u v u# -M 3 ] is any line not on any vertex of the triangle of reference, and we place u = uju^ and v == uju y the numbers [^, v] are the nonhomogeneous coordinates of the line [u v u& J referred to two of the vertices of the triangle of reference 178 COORDINATE SYSTEMS [CHAP. YII as CT-eenter and F-center respectively, and in which the line [1, 1, 1] is the line [1, 1]. If, now, we superpose these two systems of nonhom- ogeneous coordinates in the way described in the preceding section, the condition that the point (x, y) be on the line [u 9 v] is that the relation uz + vy + l = Q be satisfied (Theorem 1). It is now easy to recognize the resulting relation between the systems of homogeneous coordinates with which we started. Clearly the point (0, 1, 0) = U is the *7-center, (1, 0, 0) = F is the F-center, and (0, 0, 1) = is the third FIG. 83 vertex of the triangle of reference in the homogeneous system of line coordinates. Also the line whose points satisfy the relation # x = is the line [1, 0, 0], the line for which # s = is the line [0, 1, 0], and the line for which # 3 = is the line [0, 0, 1]. Finally, the line [1, 1] = [1, 1, 1], whose equation in nonhomogeneous coordinates is + y + 1 = 0, meets the line- x^ = in the point (0, 1, 1), and the line # 2 = in the point ( 1, 0, 1). The two coordinate systems are then completely determined (fig. 83). It now follows at once from the result of the preceding section that the condition that (x v x z , # 3 ) be on the line [u v u^ u z ] is ufli 4- 0, if none of the coordinates x lt # 2 , # 8 , u v G3] HOMOGENEOUS COORDINATES 179 is zero. To see that the same condition holds also when one (or more) of the coordinates is zero, we note first that the points (0, 1, 1), ( 1, 0, 1), and ( 1, 1, 0) are collinear. They are, in fact (fig. 83), on the axis of perspectivity of the two perspective triangles OTFand la-lylc, the center of perspectivity being J, It is now clear that the line [1, 0, 0] passes through the point (0, 1, 0;, the line [0, 1, 0] passes through the point (1, 0, 0), the line [1, 1, 0] passes through the point ( 1, 1, 0). There is thus an involution between the points (x v ;<? 2 , 0) of the line ac 9 = and the traces (j?/, %.!, 0) of the lines with the same coordinates, and this involution is given by the equations In other words, the line \u v u, 0] passes through the point ( u z) it v 0). Any other point of this line (except (0, 0, 1)) has, by definition, the coordinates ( ^ 3 , u v x s ). Hence all points (x^ # 2 , # 3 ) of the line [u v u z , 0] satisfy the relation 7^ 4- ?M*2 + Vs 0- The same argu- ment applied when any one of the other coordinates is zero estab- lishes this condition for all cases. A system of point and a system of line coordinates, when placed in the relation described above, will be said to form a system of homogeneous point and line coordinates in the plane. The result obtained may then be stated as follows : THEOREM 3. In a system of homogeneous point and line coordinates in a plane the necessary and sufficient condition that a point (% v # 2 , .T 3 ) le on a line [u v 2 , ^ 3 ] is that the relation u^ + 11^ 4- u^ = le satisfied. COROLLAEY. The equation of a line through the origin of a system of nonhomogeneous coordinates is of the form mx 4- ny = 0. EXERCISES 1. The line [1, 1, 1] is the polar of the point (1, 1, 1) with regard to the triangle of reference (cf. p. 46). 2. The same point is represented by (a l9 a 2 , a s ) and (& r 5 2 , 5 8 ) if and only if the two-rowed determinants of the matrix (J 1 ^ 2 ^ 3 ) are all zero. V*l &2 V 3. Describe nonhomogeneous and homogeneous systems of line and plane coordinates in a bundle by dualizing in space the preceding discussion. In such a bundle what is the condition that a line be on a plane ? 180 COORDINATE SYSTEMS [CHAP. Til 64. The line on two points. The point on two lines. Given two points, A = (a v a z , a,) and B = (b v & 2 , & s ), the question now arises as to what are the coordinates of the line joining them; and the dual of this problem, namely, given two lines, m = [m v m 2 , m s ] and n = [n v n n , '7& 8 ], to find the coordinates of the point of intersection of the two lines. THEOREM 4. The equation of THEOREM 4'. The equation of the line joining the points (a^a^a^) the point of intersection of the and (b v 5 2> 6 3 ) is lines [m x ,m 2 ,m 8 ] and [n v n^n s ] is = 0. Proof. "When these determinants are expanded, we get 3 =0, ni respectively. The one above is the equation of a line, the one below the equation of a point. Moreover, the determinants above both evidently vanish when the variable coordinates are replaced by the coordinates of the given elements. The expanded form just given leads at once to the following: COROLLAET V. The coordinates of the point of intersection of the lines [m v m 2 , m s ], [n v n 2 , w 8 ] are COROLLAEY 1. The coordinates of the line joining the points (a v 2 , a,), (b v 6 2 , b s ) are There also follows immediately from this theorem: COROLLARY 2. The condition COROLLARY 2'. The condition that three points A, B, C be col- that three lines m, n, p be con- linear is current is = 0. . -0. EXAMPLE. Let us verify the theorem of Desargues (Theorem 1, Chap. II) analytically. Choose one of the two perspective triangles as triangle of refer- ence, say A' = (0, 0, 1), J5 7 = (0, 1, 0), C" = (1, 0, 0), and let the center of per- spectivity be P = (1, 1, 1). If the other triangle is ABC, we may place 64,65] PEOJECTITE PENCILS 181 A = (1, 1, rt), B = (1, ft, 1), C = (c, 1, 1) : for the equation of the line PA' is x l j: 2 = ; and since J. is, by hypothesis, on this line, its first two coordi- nates must be equal, and may therefore be assumed equal to 1 ; the third coordinate is arbitrary. Similarly for the other point*. Now, from the above theorems and their corollaries we readily obtain in succession the following : The coordinates of the line A'& are [1, 0, 0]. The coordinates of the line AB are [1 ah, a 1, & 1]. Hence the coordinates of their intersection C" are C"'=(0. l -&,,i-l). Similarly, we find the coordinates of the intersection A" of the lines B'C\ BC tobe J"=(l-e,6-l,0): and, finally, the coordinates of the intersection B" of the lines C'A'* CA to be 3"= (e- 1,0,1 -a). The points A", B'\ C" are readily seen to satisfy the condition for collinearity. EXERCISES 1. Work through the dual of the example just given, choosing the sides of one of the triangles and the axis of perspectivity as the fundamental lines of the system of coordinates. Show that the work may be made identical, step for step, with that above, except for the interpretation of the symbols. 2. Show that the system of coordinates may be so chosen that a quadrangle- quadrilateral configuration is represented by all the sets of coordinates that can be formed from the numbers and 1. Dualize. 3. Derive the equation of the polar line of any point with regard to the triangle of reference. Dualize. 65. Pencils of points and lines.. Projectivity. A convenient ana- lytic representation of the points of a pencil of points or the lines of a pencil of lines is given by the following dual theorems : THEOREM 5. Any point of a THEOREM 5'. Any line of a pencil of points may le repre- pencil of lines may "be represented sented ly ly P = (X 2 a 1 + \l v X 2 a 3 + \l z , p = [|* a m 1 where A = (a x & 2 , a s ) and B = where m = [m v m 2 , m g ] and n = (&!> 5 2 , 6 S ) are any two distinct [n v n, raj are any two distinct points of the pencil. lines of the pencil. Proof. We may confine ourselves to the proof of the theorem on the left. By Theorem 4, Cor. 2, any point (x v x^ # s ) of the pencil of points on the line AB satisfies the relation 182 COORDINATE SYSTEMS [CHAP.VII \ x i x * x s\ (1) fl x s ,=0. A &s 5 a! We may then determine three numbers p, X 2 ', X/, such that we have (2) p* 1 = \'a 1 +Xft- (* = 1,2,3) The number p cannot be under the hypothesis, for then we should have from (2) the proportion x : a a : 3 = 5^ S 3 : 5 8 , which would imply that the points A and B coincide. We may therefore divide by p. Denoting the ratios \ f /p and \[/p by X 2 and \, we see that every point of the pencil may be represented in the manner specified. Conversely, every point whose coordinates are of the form specified clearly satisfies relation (1) and is therefore a point of the pencil. The points A and B in the above representation are called the "base points of this so-called parametric representation of the elements of a pencil of points. Evidently any two distinct points may be chosen as base points in such a representation. The ratio \/\ is called the parameter of the point it determines. It is here written in homoge- neous form, which gives the point A for the value \=0 and the point B for the value X 2 = 0. In many cases, however, it is more convenient to write this parameter in nonhomogeneous form, P = (a, + 7J> V a 2 + \h, a, + X6 3 ), which is obtained from the preceding by dividing by X 2 and replacing \/\ by ^- I n this representation the point B corresponds to the value X = oc. "We may also speak of any point of the pencil under this representation as the point X t : X 2 or the point X when it corre- sponds to the value X X /X 2 = X of the parameter. Similar remarks and the corresponding terminology apply, of course, to the parametric representation of the lines of a flat pencil. It is sometimes convenient, moreover, to adopt the notation A -f- XJ? to denote any point "of the pencil whose base points are A, B or to denote the pencil itself ; also, to use the notation m, + pn to denote the pencil of lines or any line of this pencil whose base lines are m, n. In order to derive an analytic representation of a projectivity between two one-dimensional primitive forms in the plane, we seek first the condition that the point X of a pencil of points A + \B be on the line p of a pencil of lines m + pn. By Theorem 3 the condition that the point X be on the line /t is the relation 65] PKOJECTIYE PENCILS 183 When expanded this relation gives t=3 i=3 l a l = 0. This is a bilinear equation whose coefficients depend only on the coor- dinates of the base points and base lines of the two pencils and not on the individual points for which the condition is sought. Placing this equation becomes 6>X + Dp. -4X B = 0, which may also be written* * The result may be stated as follows: Any perspective relation "between two one-dimensional primitive forms of different "kinds is obtained by establishing a protective correspondence between the parameters of the two forms. Since any protective correspondence between two one- dimensional primitive forms is obtained as the resultant of a sequence of such perspectivities, and since the resultant of any two linear frac- tional transformations of type (1) is a transformation of the same type, we have the following theorem : THEOREM 6. Any projectile correspondence "between two one-dimen- sional primitive forms in the plane is obtained "by establishing a protective relation between the parameters p> X of the two forms. In particular we have COROLLARY 1. Any projectimty in a one-dimensional primitive form in the plane is given by a relation of the form where X is the parameter of the form, ^ pi does not vanish because the correspondence between A ana /i is (1,1). 184 COOBDISrATE SYSTEMS [CHAP.TII COROLLARY 2. If X 1? X 2 , X 3 , \ are the parameters of four elements A v A*, A 3 , A of a one-dimensional primitive form,) the cross ratio B (A^A^ A S A 4 ) is given ly A projectivity between two different one-dimensional forms may be represented in a particularly simple form by a judicious choice of the base elements of the parametric representation. To fix ideas, let us take the case of two projective pencils of points. Choose any two distinct points A 9 B of the first pencil to be the base points, and let the homologous points of the second pencil be base points of the latter. Then to the values X = and X = oo of the first pencil must correspond the values p = and \L = oo respectively of the second. In this case the relation of Theorem 6, however, assumes the form ju, = &X. Hence, since the same argument applies to any distinct forms, we have COEOLLARY 3. If tico distinct projective one-dimensional primitive forms in the plane are represented parametrically so that the base elements form two homologous pairs, the projeetivity is represented by a relation of the form p = k\ between the parameters p, X of the two forms. This relation may be still further simplified. Taking again the case discussed above of two projective pencils of points, we have seen that, in general, to the point (a^ + l v a 2 + J a , a s + J 8 ), ie. to X = 1, corre- sponds the point (0/ + &J/, tf 2 '-f-&& 2 ', #/+&&), ie. the point /-& = . Since the point !?'=(&/, J/, 6 g ') is also represented by the set of coordi- nates (&/, &/, &/), it follows that if we choose the latter values for the coordinates of the base point B 1 , to the value X = 1 will correspond the value fi = 1, and hence we have always p. = X. In other words, we have COROLLARY 4. If two distinct one-dimensional forms are protective, the lase elements may le so chosen that the parameters of any two homologous elements are equal. Before closing this section it seems desirable to call attention explicitly to the forms of the equation of any line of a pencil and of the equation of any point of a pencil which is implied by Theorem 5 ; and Theorem 5 respectively. If we place m = m^ + w 2 # 2 H- m B x s and 65, 66] EQUATION OF A COXIC 185 71 = 71^+ nx 2 + }2 3 c>: 3 , it follows from the first theorem mentioned that the equation of any line of the pencil whose center is the inter- section of the lines m = 0, n = is given by an equation of the form ni + fin = 0. Similarly, the equation of any point of the line joining A. = a l u l + 2 3 + 3 3 = and B = J^ + 6 a w a -j- Zu? s = is of the form A +\B = 0. 66. The equation of a conic. The results of 65 lead readily to the equation of a conic. By this is meant an equation in point (line) coordinates which is satisfied by all the points (lines) of a conic, and by no others. To derive this equation, let A 9 B Le two distinct points on a conic, and let tn = m^i^-r wzA' a -t- ?/u,c = 0, (1) d = w^ + M tf v 2 + /v' 3 = 0, be the equations of the tangent at A, the tangent at J?, and the line A3 respectively. The conic is then generated as a point locus by two protective pencils of lines at A and B, in which m, p at A are homologous with jp, n at B respectively. This projectivity between the pencils 7 m + \2 } - > ^ ; p + IAH Q is given (Theorem 6, Cor. 3) by a relation (3) /A = &X between the parameters /*, X of the two pencils. To obtain the equa- tion which is satisfied by all the points of intersection of pairs of homologous lines of these pencils, and by no others, we need simply eliminate ^, X between the last three relations. The result of this elimination is (4) /-#wztt=0, which is the equation required. By multiplying the coordinates of one of the lines by a constant we may make k = 1. Conversely, it is obvious that the points which satisfy any equation of type (4) are the points of intersection of homologous lines in the pencils (2), provided that /z, = k\. If m, n, p are fixed, the condition that the conic (4) shall pass through a point (a v a v a z ) is a linear equation in Je. Hence we have 186 COOEDIKATE SYSTEMS [CHAP. VII THEOEEM 7. If m = 0, n = 0, p = are the equations of two distinct tangents of a conic and the line joining (heir points of con- tact -respectively, the point equa- tion of the conic is of the form p~ kmn = 0. The coefficient "k is determined by any third point on the conic. Con- versely, the points which satisfy an equation of the above form constitute a conic of which m = and n = are tangents at points on p = 0. COROLLAEY. By properly choos- ing the triangle of reference, the point equation of any conic may te put in the form THBOEEM 7.' If A = 0, B = 0, C = are the equations of two distinct points of a conic and the intersection of the tangents at these points respectively, the line equa- tion of the conic is of the form where r^= 0, # 3 = are two tan- gents, and # 2 = is the line join- ing their points of contact. The coefficient k is determined ly any third line of the conic. Con- versely, the lines which satisfy an equation of the above form consti- tute a conic of which A = and B = are points of contact of the tangents through C = 0. COEOLLAEY. By properly choos- ing the triangle of reference, the line equation of any conic may be put in the form w 2 2 2^3 = 0, where ^= 0, % 3 = are two points, and u 2 = is the intersection of the tangents at these points. It is clear that if we choose the point (1, 1, 1) on the conic, we have k = 1. Supposing the choice to have been thus made, we inquire regarding the condition that a line [u v u z , u^ be tangent to the conic This condition is equivalent to the condition that the line whose equation is ^a^-f- ^ 2 # 2 + U B % & = shall have one and only one point in common with the conic. Elimi- nating # 8 between this equation and that of the conic, the points common to the line and the conic are determined by the equation UjSB? + Ufl^ + 1(, 3 %j = 0. The roots of this equation are equal, if and only if we have u% 4 u^ = 0. tK5,C7] LIXEAE TEAXSFOP^IATIOXS 1ST Since this is the line equation of all tangents to the conic, and since it is of the form given in Theorem 7', Cor., above, we have here a new proof of the fact that the tangents to a point conic form a li/ic conic (cf. Theorem 11, Chap. V). When the linear expressions for M, n, p are substituted in the equa- tion j> 2 ~kmn = of any conic, there results, when multiplied out, a homogeneous equation of the second degree in x v # 2 , tt i s , which may be written in the form (1) a n x{ + a 22 ,<7 + a^c* + 2 a 12 ^ 2 + 2 a^c^ + 2 a^x^ = 0. We have seen that the equation of every conic is of this form. We have not shown that every equation of this form represents a conic (see 85, Chap. IX). EXERCISE Shcrw that the conic degenerates into (distinct or coincident) straight lines, if and only if we have Dualize. (A, E, P, H ) 67. Linear transformations in a plane. We inquire now concern- ing the geometric properties of a linear transformation (1) Such a transformation transforms any point (x v # 2 , # 3 ) of the plane into a unique point (#/, # 2 ; , x) of the plane. Eeciprocally, to every point x 1 will correspond a unique point x, provided the determinant of the transformation A ^31 ^82 a X is not 0. For we may then solve equations (1) for the ratios x^ix^n in terms of a?/: x : x% as follows : (2) />', p'x, 188 COORDINATE SYSTEMS [CHAP.VII here the coefficients A tJ are the cofactors of the elements a t) respec- tively in the determinant A. Further, equations (1) transform every line in the plane into a unique line. In fact, the points x satisfying the equation UjJL\ + M 2 # 3 + tt a # 8 = are, by reference to equations (2), transformed into points x 1 satisfy- ing the equation (A u ii 1 + A^IS + A^ x[ 4- (A^ + A^u^ + A^u t ) ^ + (A^U! + A^u s + A 5Q u s ) 3 ' = 0, which is the equation of a line. If the coordinates of this new line be denoted by [/, u^ u\ 9 we clearly have the following relations between the coordinates [K V u^ w s ] of any line and the coordinates [/, u^, u\ of the line into which it is transformed by (1): (3) We have seen thus far that (1) represents a collineation in the plane in point coordinates. The equations (3) represent the same collineation in line coordinates. It is readily seen, finally, that this collineation is protective. For this purpose it is only necessary to show that it transforms any pencil of lines into a projective pencil of lines. But it is clear that if m = and n are the equations of any two lines, and if (1) trans- forms them respectively into the lines whose equations are m 1 = and ?2/=0, any line ??i + Xft=0 is transformed into m f +\n f Q, and the correspondence thus established between the lines of the pencils has been shown to be projective (Theorem 6). Having shown that every transformation (1) represents a projective collraeation, we will now show conversely that every projective collineation in a plane may be represented by equations of the form (1). To this end we recall that every such collineation is completely determined as soon as the homologous elements of any complete quadrangle are assigned (Theorem 18, Chap. IV). If we can show that likewise there is one and only one transformation of the form (1) changing a given quadrangle into a given quadrangle, it will follow that, since the linear transformation is a projective collineation, it is the given projective collineation. 67] LINEAR TRANSFORiEATIOXS 189 Given any protective collineation in a plane, let the fundamental points (0, 0, 1), (0, 1, 0), (1, 0, 0), and (1, 1, 1) of the plane (which form a quadrangle) be transformed respectively into the points A - (a v a v aj, B = (b v 6 2 , J s ), C = (c v c z , ), and I) = (d v tl, tl), form- ing a quadrangle. Suppose, now, we seek to determine the coefficients of a transformation (1) so as to effect the correspondences just indi- cated. Clearly, if (0, 0, 1) is to be transformed into (a v a, ,), we must have . a ls =\a v r/ 23 = X 2 , a 3G =X s , X being an arbitrary factor of proportionality, the value (=?= 0) of which we may choose at pleasure. Similarly, we obtain Since, by hypothesis, the three points A, B, C are not collinear, it follows from these equations and the condition of Theorem 4, Cor. 2, that the determinant A of a transformation determined in this way is not 0. Substituting the values thus obtained in (1), it is seen that if the point (1, 1, 1) is to be transformed into (d v d^ d s ), the following relations must hold : /w7 1 =<r pel* = c Placing p = 1 and solving this system of equations for v 9 p, \ we obtain the coefficients a tj of the transformation. This solution is unique, since the determinant of the system is not zero. Moreover, none of the values X, p, v will be ; for the supposition that v = 0, for example, would imply the vanishing of the determinant d l \ which in turn would imply that the three points D, B, A are collinear, contrary to the hypothesis that the four points A, B> C 9 D form a complete quadrangle. Collecting the results of this section, we have THEOREM 8. Any protective colKneatwn in the plane may be repre- sented in point coordinates Tyy equations of form (1) or in line coordi- nates ly equations of form (3), and in each case the determinant of 190 COOBDINATE SYSTEMS [CHAP.VII the transformation is different from ; conversely, any transforma- tion of one of these forms in which the determinant is different from represents a protective collineation in the plane. COROLLARY 1. In nonhomogeneous point coordinates the equations of a projectile collineation are COROLLARY 2. If the singular line of the system of nonhomogeneous point coordinates is transformed into itself, these equations can 5e written x ! = a* 6, =0. 68. Collineations between two different planes. The analytic form of a collineation between two different planes is now readily derived. Let the two planes be a and /3, and let a system of coordinates be established in each, the point coordinates in a being (x v % 2 , # 3 ) and the point coordinates in y8 being (y v y z , y s ). Further, let the isomor- phism between the number systems in the two planes be established in such a way that the correspondence established by the equations Vi** *v y =a fc y.^afe is protective. It then follows, by an argument (cf. 59, p. 166), which need not be repeated here, that any collineation between the two planes may be obtained as the resultant of a projectivity in the plane a, which transforms a point X, say, into a point X\ and the pro- jectivity r = X f between the two planes. The analytic form of any protective collineation letween the two planes is therefore : with the determinant A of the coefficients 'different from 0. And, con- versely, every such transformation in which A = represents a projec- tive collineation letween the two planes. 69. Nonhomogeneous coordinates in space. Point coordinates in space are introduced in a way entirely analogous to that used for the introduction of point coordinates in the plane. Choose a tetrahedron of reference OUVW and label the vertices = 0^= O y = 0,,, Z7= oo a , COORDINATES 1ST SPACE 191 F= oo y , JT= ex), (fig. 84) ; and on the lines O^oo,, O^oo,, O s co 2 , called respectively the x-axis, the y-axis, the 2-axis, establish three scales by choosing the points 1 X9 l y , 1 2 . The planes Ooo a co y , Ooo x co g9 Ott y ao a are called the xy-plane, xz-plane, yz-plane respectively. The point is called the origin. If P is any point not on the plane co x oo y oo z> which is called the singular plane of the coordinate system, the plane P 00^003 meets the #-axis in a point whose nonhomogeneons coordinate in the scale (0^, 1^, co x ) we call a. Similarly, let the plane meet the 2/-axis in a point whose nonhomogeneous coordinate in the scale (0,, l y) co y ) is I ; and let the plane Poo x co y meet the z-axis in a point whose nonhomogeneous coordi- nate in the scale (O a , 1 Z9 <x> z ) is c. The numbers a, &, c are then the nonhomo- geneous #-, y- 9 and z-coor- dinates of the point P. Conversely, any three numbers a 9 1 9 c determine three points A 9 B 9 C on the x- 9 y-> and #-axes respectively, and the three planes AaOyOo^ B<x> x <x> a , Cco x <x> y meet in a point P whose coordinates are a 9 6, c. Thus every point not on the singular plane of the coordinate system determines and is determined by three coordinates. The point P is then repre- sented by the symbol (a 9 &, c). The dual process gives rise to the coordinates of a plane. Point and plane coordinates may then be put into a convenient relation, as was done in the case of point and line coordinates in the plane, thus giving rise to a system of simultaneous point and plane coordinates in space. We will describe the system of plane coordinates with reference to this relation. Given the system of nonhomogeneous point coordinates described above, establish in each of the pencils of planes on the lines VW 9 UW, UV a scale by choosing the plane UVW as the zero plane M = O r = O w in each of the scales, and letting the planes VW 9 UW 9 UV be the planes oo tt , oo , oo^ respectively. In the ^scale FIG. 84 192 COORDINATE SYSTEMS [CHAP.VH let that plane through. 7W be the plane l u , which meets the aj-axis in the point l a . Similarly, let the plane l v meet the ?/-axis in the point l y ; and let the plane l w meet the z-axis in the point !. The swscale, inscale, and w-scale being now completely determined, any plane TT not on the point (which is called the singular point of this system of plane coordinates) meets the #-, y-, and #-axes in three points L, M> N which determine in the u-, v-, and ^-scales planes whose coordinates, let us say, are l } m, n. These three numbers are called the nonhomogeneous plane coordinates of TT. They completely determine and are completely determined by the plane TT. The plane TT is then denoted by the symbol [I, m, n]. In this system of coordinates it is now readily seen that the con- dition that the point (a, 5, c) be on the plane \l 9 m,ri\ is that the relation la + mb + nc + l = Q le satisfied. It follows readily, as in the planar case, that the plane |7, m> n] meets the #-, y-, and #-axes in points whose coordinates on these axes are 1/, 1/m, and 1/n respec- tively.* In deriving the above condition we will suppose that the plane TT = |7, m 9 n] does not contain two of the points U, F, IF, leav- ing the other case as an exercise for the reader. Suppose, then, that U= co x and F= ao y are not on TT. By projecting the ys-plane with U as center upon the plane TT, and then projecting TT with F as center on the ##-plane, we obtain the following perspectivities : where (x, y, z) represents any point on ?r. The product of these two perspectivities is a projectivity between the y^-plane and the ##-plane, by which the singular line of the former is transformed into the sin- gular line of the latter. Denoting the ^-coordinate of points in the 2/2-plane by z f , this projectivity is represented (according to Theorem 8, Cor. 2, and 68) by relations of the form We proceed to determine the coefficients a v l v c v The point of intersection of TT with the y-axis is (0, 1/m, 0), and is clearly * This statement remains valid even if one or two of the numbers Z, m, n are zero (they cannot all be zero unless the plane in question is the singular plane which we exclude from consideration), provided the negative reciprocal of be denoted by the symbol o>. 09] COOBDINATES DT SPACE 193 transformed by the projectivity in question into the point (0, 0, 0). Hence (1) gives CI= -,IT The point of intersection of TT with the z-axis is, if w=0, (0, 0, 1/n) and is transformed into itself. Hence (1) gives n m 7 M or 5. = --- m If n = 0, we have at once \ = 0. Finally, the point of intersection of TT with the #-axis is ( 1/7, 0, 0), and the transform of the point (0, 0, 0). Hence we have or a, = --- 1 m Hence (1) becomes y = -- x -- z -- > m m m a relation which must be satisfied by the coordinates (x, y, z) of any point on TT. This relation is equivalent to Ix + my 4- nz -j- 1 = 0. Hence (a, 6, c) is on [7, m,n], if (2) Za + m& + nc + 1 = 0. Conversely, if (2) is satisfied by a point (a, 6, c), the point (0, 5, e) = P is transformed by the projectivity above into (#, 0, c) = $, and hence the lines JP 27 and Q V which meet in (, &, c) meet on TT. DEFINITION. An equation which DEFINITION. An equation which is satisfied by all the points (x, y, ) is satisfied by all the planes [u,v,w] of a plane and by no other points on a point and by no other planes is called the point equation of the is called the plane equation of the plane. point. The result of the preceding discussion may then be stated as follows : THEOREM 9. The point equation THEOREM 9'. The plane equation of the plane [I, m, n] is of the point (a, I, c) is = Q. au + bv + cw + 1 = 0. 194 COORDINATE SYSTEMS [OHAP.VII 70. Homogeneous coordinates in space. Assign to the vertices 0, U, V, W of any tetrahedron of reference the symbols (0, 0, 0, 1), (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0) respectively, and assign to any fifth point T not on a face of this tetrahedron the symbol (1, 1, 1, 1). The five points 0, U 9 V, W, T are called the frame of reference of the system of homogeneous coordinates now to be described. The four lines join- ing T to the points 0, U, V, W meet the opposite faces in four points, which we denote respectively by (1, 1, 1, 0), (0, 1, 1, 1), (1, 0, 1, 1), (1, 1, 0, 1). The planar four-point (0, 0, 0, 1), (0, 0, 1, 0), (0, 1, 0, 0), (0, 1, 1, 1) we regard as the frame of reference (0, 0, 1), (0, 1, TO), (1, 0, 0), (1, 1, 1) of a system of homogeneous coordinates in the plane. To any point in this plane we assign the coordinates (0, # 2 , # 3 , # 4 ), if its coordinates in the planar system just indicated are (# 2 , # 3 , # 4 ). In like manner, to the points of the other three faces of the tetrahedron of reference we assign coordinates of the forms (x v 0, # 3 , # 4 ), (x v x 2t 0, # 4 ), and (x v x z> x s , 0). The coordinates of the points in the faces opposite the vertices (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1) satisfy respec- tively the equations x^ = 0, # 2 = 0, x z = 0, x = 0. To the points of each edge of the tetrahedron of reference a notation has been assigned corresponding to each of the two faces which meet in the edge. Consider, for example, the line of intersection of the planes x l =Q and # 2 = 0. Eegarding this edge as a line of x t =^ 0, the coordinate system on the edge has as its fundamental points (0, 0, 1, 0), (0, 0, 0, 1), (0, 0, 1, 1). The first two of these are vertices of the tetra- hedron of reference, and the third is the trace of the line joining (0, 1, 0, 0) to (0, 1, 1, 1). On the other hand, regarding this edge as a line of # 2 =0, the coordinate system has the vertices (0, 0, 1, 0) and (0, 0, 0, 1) as two fundamental points, and has as (0, 0, 1, 1) the trace of the line joining (1, 0, 0, 0) to (1, 0, 1, 1). But by construction the plane (0, 1, 0, 0)(1, 0, 0, 0)(1, 1, 1, 1) contains both (0, 1, 1, 1) and (1, 0, 1, 1), so that the two determinations of (0, 0, 1, 1) are identical. Hence the symbols denoting points in the two planes x l = and # 2 =0 are identical along their line of intersection. A similar result holds for the other edges of the tetrahedron of reference. THEOREM 10, DEFINITION. If P is any point not on a face of the tetrahedron of reference, there exist four numbers x v # 2 , # 3 , x^ all different from zero, suck that the projections of P from the four vertices (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1) respectively upon their 70] COORDINATES IN SPACE 195 opposite faces are (0, <e a , x v xj, (x v 0, X B) a?J, (^ re s , 0, # 4 ), (x v x 2 , x z , 0). l^ese four numbers are called the homogeneous coordinates of P and P is denoted ly (x v x 2 , X B , a?J. Any ordered set of four members, not all zero, determine uniquely a point in space whose coordinates they are. Proof. The line joining P to (1, 0, 0, 0) meets the opposite face in a point (0, x z , x 2 , # 4 ), which is not an edge of the tetrahedron of refer- ence, and such therefore that none of the numbers # 2 , X B , x 4 is zero. Likewise the line joining P to (0, 1, 0, 0) meets the opposite face in a point (as/, 0, # 8 ', # 4 ), such that none of the numbers x[, # 8 ', x is zero. But the plane P(l, 0, 0, 0) (0, 1, 0, 0) meets x l = in the line joining (0, 1, 0, 0) to (0, x 2 , # 3 , # 4 ), and meets x 2 = in the line joining (1, 0, 0, 0) to (as/, 0, # 3 ', xl). By the analytic methods already devel- oped for the plane, the first of these lines meets the edge common to x 1 = and x 2 = in the point (0, 0, a? 8 , # 4 ), and the second meets it in the point (0, 0, #/, xl). But the points (0, 0, a? 8 , # 4 ) and (0, 0, # 8 ', x^) are identical, and hence, by the preceding paragraph, we have # s /# 4 = %l/%l* Hence, if we place x l = x^xjxl, the point (as/, 0, x, xl) is identical with (x v 0, x 9 , x). The line joining P to (0, 0, 1, 0) meets the face # 8 = in a point (x? 9 x, 0, x^. By the same reasoning as that above it follows that we have x/xl r = xjx and %2/xl 1 ttj%# so that the point (as/', x^ 0, x^) is identical with (x v # 2 , 0, a? 4 ). Finally, the line joining P to (0, 0, 0, 1) meets the face # 4 = in a point which a like argument shows to be (x v x 2 , X B , 0). Conversely, if the coordinates (x v # 2 , a? 3 , # 4 ) are given, and one of them is zero, they determine a point on a face of the tetrahedron of reference. If none of them is zero, the lines joining (1, 0, 0, 0) to (0, x z9 x 39 # 4 ) and (0, 1, 0, 0) to (x v 0, # 8 , # 4 ) are in the plane (1, 0, 0, 0) (0, 1, 0, 0) (0, 0, x s) x 4 ), and hence meet in a point which, by the reasoning above, has the coordinates (x v x z , # 8 , # 4 ). COEOLLAEY. The notations (x^x^x^x^) and (kx v Jcx 2 , Jcx B , kx^) denote the same point for any value of 7c not equal to zero. Homogeneous plane coordinates in space arise by the dual of the above process. The four faces of a tetrahedron of reference are denoted respectively by [1, 0, 0, 0], t [0, 1, 0, 0], [0, 0, 1, 0], and [0, 0, 0, 1]. These, together with any plane [1, 1, 1, 1] not on a vertex of the tetrahedron, form the frame of reference. The four lines of inter- section of the plane [1, 1, 1, 1] with the other four planes in the order 196 COOKDINATE SYSTEMS [CHAP.VII above are projected from the opposite vertices by planes which are denoted by [0,1,1,1], [1,0,1,1], [1,1,0,1], [1,1,1,0] respectively. The four planes [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1], and [0, 1, 1, 1] form, if the first in each of these symbols is suppressed, the frame of reference of a system of homogeneous coordinates in a bundle (the space dual of such a system in a plane). The center of this bundle is the vertex of the tetrahedron of reference opposite to [1, 0, 0, 0]. To any plane on this point is assigned the notation [0, u v u# ^J, if its coordinates in the bundle are [u z , u s> wj. In like manner, to the planes on the other vertices are assigned coordinates of the forms [u v 0, u s , u], [u v u v 0, ttj, [u v u 2 , u 3 , 0]. The space dual of the last theorem then gives : THEOKEM 10'. DEFINITION. Ifir is any plane not on a vertex of the tetrahedron of reference, there exist four numbers u v u^ u 9) u *> a ^ differ- ent from zero, such that the traces of IT on the four faces [1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1] respectively are projected from the opposite vertices "by the planes [0, u^, u %9 wj, [u v 0, w s , wj, [u v u z) 0, % 4 ], \u v u z ,u 2 , 0], These four members are called the homogeneous coordinates 0/7T, and TT is denoted ly [u v u z , u s , u\. Any ordered set of four num- lers, not all zero, determine uniquely a plane whose coordinates they are. By placing these systems of point and plane coordinates in a proper relation we may now readily derive the necessary and sufficient con- dition that a point (x v a? a , x s> a? 4 ) be on a plane \u v u%, u &) u^\. This condition will turn out to be We note first that in a system of point coordinates as described above the sis points (- 1, 1, 0, 0), (- 1, 0, 1, 0), (- 1, 0, 0, 1), (0, - 1, 1, 0), (0, 0, 1, 1), (0, 1, 0, 1) are coplanar, each being the harmonic con- jugate, with respect to two vertices of the tetrahedron of reference, of the point into which (1, 1, 1, 1) is projected by the line joining the other two vertices. The plane containing these is, in fact, the polar of (1, 1, 1, 1) with respect to the tetrahedron of reference (cf. Ez. 3, p. 47). Now choose as the plane [1, 0, 0, 0] the plane ^= 0, as the plane [0, 1, 0, 0] the plane # 2 = 0, as the plane [0, 0, 1, 0] the plane x s = 0, as the plane [0, 0, 0, 1] the plane x 4 = 0, 70] COORDINATES IN SPACE 197 as the plane [1, 1, 1, 1] the plane containing the points ( 1, 1, 0, 0), (-1,0,1,0), (-1,0,0,1)- With this choice of coordinates the planes [1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], and [1, 1, 1, 0] through the vertex F 4 , say, whose point coordinates are (0, 0, 0, 1), meet the opposite face x = in lines whose equations in that plane are Hence the first three coordinates of any plane [u v ^ 2 , u v 0] on F 4 are the line coordinates of its trace on # 4 = 0, in a system so chosen that the point (SG V x 2 , # 3 ) is on the line [u v u S9 u s ] if and only if the relation u^ + U Q X & + u s x s = is satisfied. Hence a point (x v x z , x v 0) lies on a plane [u l9 u 2 , u^ 0] if and only if we have ufa + UyfB 9 + ^ 3 # s = 0. But any point (sc v # 2 , # 8 , # 4 ) on the plane [u v u^ u^ 0] has, by definition, its first three coordinates identical with the first three coordinates of some point oa the trace of this plane with the plane x = 0. Hence any point (x 19 x 2 , x 3 , x 4 ) on [u v u 2 , u s , 0] satisfies the condition ufa + u^x z + u z x s + ufa*= 0. Applying this reasoning to each of the four vertices of the tetrahedron of reference and dualizing, we find that if one coordinate of \u v u 2 , ^ 3 , %J is zero, the necessary and sufficient condition that this plane contain a point (x v % 2 , # 8 , x 4 ) is that the relation U^ + U& + Ufa + Ufa = le satisfied; and if one coordinate of (x v x z , X B , x 4 ) is zero, the neces- sary and sufficient condition that this point be on the plane [^ v u &9 u^u^\ is likewise that tJie relation just given "be satisfied. Confining our attention now to points and planes no coordinate of which is zero, let xjx^x 9 xjx^y^ a? 8 /# 4 = 2, and let uju^% 9 w a /w 4 =0, u z f%^w. Since x 9 y, z are the ratios of homogeneous coordinates on the lines # 2 = # 3 = 0, x^ = x 3 = 0, and x = x% = respec- tively, they satisfy the definition of nonhomogeneous coordinates given in 69. And since the homogeneous coordinates have been so chosen that the plane (u v % z , u^ u^ meets the line x z = x z = in the point ( u 9 0, 0, u^ = ( 1/u, 0, 0, 1), it follows that u, v } w are nonhomogeneous plane coordinates so chosen that a point (x, y, z) 9 none of whose coordinates is zero, is on a plane [%, v> w\ none of whose coordinates is zero, if and only if we have (Theorem 9) ux + *oy -I- ^^ + 1 = ; 198 COORDINATE SYSTEMS [CHAP. VII that is, if and only if we have U& 4- Ufa + u s x 3 + Ufa = 0. This completes for all cases the proof of THEOREM 11. The necessary and sufficient condition that a point (Xy x 2i X B) # 4 ) le on a plane [u v u^ u s> u^ is that the relation Ufa + Ufa + Ufa + Ufa = be satisfied. By methods analogous to those employed in 64 and 65 we may now derive the results of Exs. 1-8 below. EXERCISES 1. The equation of the plane through the three points A = (a v a 2 , # 3 , a 4 ), B = (1 19 b z , 3 , Z> 4 ), C = (c x , c 2 , c 3 , c 4 ) is #1 2T 5.' a 5 1 & 2 5 3 & 4 = 0. Dualize. 2. The necessary and sufficient condition that four points -4, B, C 7 D be coplanar is the vanishing of the determinant b l 5 3 b z 5 4 3. The necessary and sufficient condition that three points A, B, C be collinear is the vanishing of the three-rowed determinants of the matrix 4. Any point of a pencil of points containing A and -B may be represented by P = 5. Any plane of a pencil of planes containing m = [%, m 2 , w 8 , wi 4 ] and n = [n lf n 2 , TZ S , n 4 ] may be represented by 6. Any projectivity between two one-dimensional primitive forms (of points or planes) in space is expressed by a relation between their parameters X, p, of the form If the base elements of the pencil are homologous, this relation reduces to 70,71] LINEAE TRAITSFOBMATION 1 199 7. If \ v X 21 \ 3 , \ 4 are the parameters of four points or planes of a pencil, tlieir cross ratio is 8. Any point (plane) of a plane of points (bundle of planes) containing the noncollinear points A, B, C (planes a, /?, y) may be represented by P = (A^ + X^ + Ag^, A.ja 2 4- Xj&j -f X 3 c 2 , X^g + X 2 Z> 8 -f A 8 <? 8 , A.^ + X^ -f \ 3 c 4 ). 9. Derive the equation of the polar plane of any point with regard to the tetrahedron of reference. 10. Derive the equation of a cone. *11. Derive nonhomogeneous and homogeneous systems of coordinates in a space of four dimensions. 71. Linear transformations in space. The properties of a linear transformation in space are similar to those found in 68 for the linear transformations in a plane. If the determinant of the transformation is different from zero, the transformation (1) will have a unique in- verse, viz.: < ; where the coefi&cients ^t y . are the cof actors of the elements a y . respec- tively in the determinant A. The transformation is evidently a collineation, as it transforms the ^ G %!#! + ^ 2 # 2 + w 8 # s + i6 4 a; 4 = into the plane 2 4- A BB u s -h 200 COOKDINATE SYSTEMS [CHAP, vn Hence the collineation (1) produces on the planes of space the trans- formation ( ' a-iol aru[ = A^Ut 4- To show that the transformation (1) is projective consider any pencil of planes In accordance with (2) this pencil is transformed into a pencil of the form (a^x 1 + a fa + ax B + a 4 \) + X (6^ + 1 fa + I fa + 1^ = 0, and these two pencils of planes are projective (Ex. 6, p. 198). Finally, as in 67, we see that there is one and only one trans- formation (1) changing the points (0, 0, 0, 1), (0, 0, 1, 0), (0, 1, 0, 0), (1, 0, 0, 0), and (1, 1, 1, 1) into the vertices of an arbitrary complete five-point in space. Since this transformation is a projective collinea- tion, and since there is only one projective collineation transforming one five-point into another (Theorem 19, Chap. IV), it follows that every projective collineation in space may be represented by a linear transformation of the form (1). This gives THEOREM 12. Any projective collineation of space may le repre- sented in point coordinates ly equations of the form (1), or in plane coordinates by equations of the form (3). In each case the determinant of the transformation is different from zero. Conversely, any trans- formation of this form in which the determinant is different from zero represents a protective collineation of space. COEOLLAET 1. In nonhomogeneous point coordinates a projective collineation is represented ly the linear fractional equations . in which the determinant A is different from zero. 71,72] FINITE SPACES 201 COROLLARY 2. If the singular plane of the nonhomogeneous system is transformed into itself, these equations reduce to x f = a 72. Finite spaces. It will be of interest at this point to emphasize again the generality of the theory which we are developing. Since all the developments of this chapter are on the basis of Assumptions A, E, and P only, and since these assumptions imply nothing regard- ing the number system of points on a line, except that it be commu- tative, it follows that we may assume the points of a line, or, indeed, the elements of any one-dimensional form, to be in one-to-one recip- rocal correspondence with the elements of any commutative number system. We may, moreover, study our geometry entirely by analytic methods. From this point of view, any point in a plane is simply a set of three numbers (x v # 2 , # 3 ), it being understood that the sets (x v x 2> X B ) and (Jcx v Jcx 2 , Jcx s ) are equivalent for all values of A in the number system, provided k is different from 0. Any line in the plane is the set of all these points which satisfy any equation of the form 2^ + u z x 2 + u 8 x B = 0, the set of all lines being obtained by giving the coefficients (coordinates) [u v u^ u$] all possible values in the number system (except [0, 0, 0]), with the obvious agreement that [u^u^Uz] and [ku 19 ku s , JcUi] represent the same line (^=^0). By letting the number system consist of all ordinary rational numbers, or all ordinary real numbers, or all ordinary complex numbers, we obtain respectively the analytic form of ordinary rational, or real, or complex projective geometry in the plane. All of our theory thus far applies equally to each of these geometries as well as to the geometry obtained by choosing as our number system any field whatever (any ordinary algebraic field, for example). In particular, we may also choose a finite field, i.e. one which con- tains only a finite number of elements. The simplest of these are the modular fields, the modulus being any prime number^.* If we * A modular field with modulus p is obtained as follows : Two integers n, n f (positive, negative, or zero) are said to be congruent modulo p, written n=n', mod.p, if the difference n n' is divisible by p. Every integer is then congruent to one and only one of the numbers 0, 1, 2, , p 1. These numbers are taken as the elements of our field, and any number obtained from these ly addition, subtraction* 202 COORDINATE SYSTEMS [CHAP.VH consider, for example, the case p = 2, our number system contains only the elements and 1. There are then seven points, which we will label A, B, C, D, E, F, G, as follows : A = (0, 0, 1), B = (0, 1, 0), <? = (!, 0, 0), D = (0, 1, 1), J0 = (l, 1, 0), ^ = (1, 1, 1), ff (1, 0, 1). The reader will readily verify that these seven points are arranged in lines according to the table A B C D F G B C D E F G A D E F G A B C, each column constituting a line. Tor example, the line a? 1 = clearly consists of the points (0, 0, 1) = A, (0, 1, 0) = B, and (0, 1, 1) = D, these "being the only points whose first coordinate is 0. We have labeled the points of this finite plane in such a way as to exhibit clearly its abstract identity with the system of triples used for illustrative pur- poses in the Introduction, 2.* EXERCISES 1. Verify analytically that two sides of a complete quadrangle containing a diagonal point are harmonic with the other two diagonal points. 2. Show analytically that if two projective pencils of lines in a plane have a self -corresponding line, they are perspective. (This is equivalent to Assump- tion P.) 3. Show that the lines whose equations are x 1 + \x 2 = 0, x% 4- pjc 8 = 0, and #s +vx t = are concurrent if X/w = 1 ; and that they meet the opposite sides of the triangle of reference respectively in collinear points, if X/xv = 1. 4. Find the equations of the lines joining (c v c 2 , c 8 ) to the four points (1, 1, 1), and determine the cross ratios of the pencil. and multiplication, if not equal to one of these elements, is replaced by the element to which it is congruent. The modular field with modulus 5, for example, consists of the elements 0, 1, 2, 3, 4, and we have as examples of addition, subtraction, and multiplication 14-3 = 4, 2 + 3 = (since 5 = 0, mod. 5), 1 - 4 = 2, 2 . 3 = 1, etc, Furthermore, if a, 5 are any two elements of this field (a^ 0), there is a unique element x determined by the congruence ax = 6, mod. p; this element is defined as the quotient b/a (For the proof of this proposition the reader may refer to any standard text on the theory of numbers.) In the example discussed we have, for example, 4/3 = 3. * For references and a further discussion of finite projective geometries see a paper by O. Veblen and W. H. Bussey, Finite Projective Geometries, Transactions of the American Mathematical Society, Vol. VII (1906), pp. 241-259. Also a sub- sequent paper by 0. Veblen, Collineations in a Finite Projective Geometry, Trans- actions of the American Mathematical Society, Vol. VIII (1907), pp. 266-268. 72] EXERCISES 203 5. Show that the throw of lines determined on (c l9 c 2 , c s ) by the four points (1, 1, 1) is projective with (equal to) the throw of lines determined on (b 19 & 3 , & 3 ) by the points (a l9 a 2 , 3 ), if the following relations hold: a 3 = 0, ajajb? + a^bj + a^ajb} = 0, and that the sis cross ratios are a 3 /a 3 , a 9 /a 19 a^/a^ , a s /a, - aja^ 2 /i (C. A. Scott, Mod. Anal. Geom., p. 50). 6. Write the equations of transformation for the five types of planar col- lineations described in 40, Chap. IV, choosing points of the triangle of reference as fixed points. 7. Generalize Ex. 6 to space. 8. Show that the set of values of the parameter X of the pencil of lines m + \n = is isomorphic with the scale determined in this pencil by the lines for which the fundamental lines are respectively the lines X = 0, 1, oo 9. Show directly from the discussion of 61 that the points whose non- homogeneous coordinates x 9 y satisfy the equation y = x are on the line joining the origin to the point (1, 1). 10. There is then established on this line a scale whose fundamental points are respectively the origin, the point (!?!) and the point in which the line meets the line fc . The lines joining any point P in the plane to the points oo y , 003. meet the line y = x in two points whose coordinates in the scale just determined are the nonhomogeneous coordinates of P, so that any point in the plane (not on loo) is represented by a pair of points on the line y = x. Hence, show that in general the points (a;, y) of any line in the plane determine on the line y = x a projectivity with a double point on Z ; and hence that the equa- tion of any such line is of the form y = ax -f- b. What lines are exceptions to this proposition ? 11. Discuss the modular plane geometry in which the modulus is^? = 3 ; and by properly* labeling the points show that it is abstractly identical with the system of quadruples exhibited as System (2) on p. 6. 12. Show in general that the modular projective plane with modulus p contains p* + p + 1 points and the same number of lines ; and that there are p 4- 1 points (lines) on every line (point). 13. The diagonal points of a complete quadrangle in a modular plane pro- jective geometry are collinear if and only if p = 2. 14 * Show that the points and lines of a modular plane all belong to the same net of rationality. Such a plane is then properly projective without the use of Assumption P. 15. Show how to construct a modular three-space. If the modulus is 2, show that its points may be labeled 0, 1, . . . , 14 in such a way that the planes are the sets of seven obtained by cyclic permutation from the set 1 4 6 11 12 13 (i.e. 1 2 5 7 12 13 14, etc.), and that the lines are ob- tained from the lines 014, 1 2 8, 5 10 by cyclic permutations. (For a 204 GOOEDINATE SYSTEMS [CHAP.VII study of this space, see G. M, Conwell, Annals of Mathematics, Vol. 11 (1910), p. 60.) 16. Show that the ten diagonal points of a complete five-point in space (0, 0, 0, 1), (0, 0, 1, 0), (0, 1, 0, 0), (1, 0, 0, 0), (1, 1, 1, 1) are given by the remaining sets of coordinates in which occur only the digits and 1. 17. Show that the ten diagonal points in Ex. 16 determine in all 45 planes, of which each of a set of 25 contains four diagonal points, while each of the remaining 20 contains only three diagonal points. Through any diagonal point pass 16 of these planes. The diagonal lines, i.e. lines joining two diagonal points, are of two kinds : through each of the diagonal lines of the first kind pass five diagonal planes ; through each line of the second kind pass four diagonal planes. 18. Show how the results of Ex. 17 are modified in a modular space with modulus 2 ; with modulus 3. Show that in the modular space with modulus 5 the results of Ex. 17 hold without modification. * 19. Derive homogeneous and nonhomogeneous coordinate systems for a space of n dimensions, and establish the formulas for an n-dimensional projective collineation. CHAPTER VIII PROJECTTVITIES IN ONE-DIMENSIONAL FORMS* 73. Characteristic throw and cross ratio. THEOREM 1. If M 9 N are double points of a projectivity on a line, and AA f , BB f are any two pairs of homologous points (i.e. if MNAB -^ MNA'ff), then MNAA -^ MNBB 1 . Proof. Let S 9 S f be any two distinct points on a line through M (fig. 85), and let the lines SA and S f A r meet in A" 9 and SB and S N S'B' meet in B". The points A", B", Nsxe then collinear (Theorem 23, Chap. IV). If the line A n B" meets SS' in a point Q> we have A" B" MNAA f = MQ8S 1 = MNBB f . A A This proves the theorem, which may also be stated as follows : The throws consisting of the pair of double points in a given order and any pair of homologous points are all equal. DEFINITION. The throw "T(MN 9 AA 1 ), consisting of the double points and a pair of homologous points of a projectivity, is called the charac- teristic throw of the projectivity ; and the cross ratio of this throw is called the characteristic cross ratio of the projectivity. | * All the developments of this chapter are on the basis of Assumptions A, E, P, H . t Since the double points enter symmetrically, the throws T (MN^ AA 7 ) and T(JO(f, AA") may be used equally well for the characteristic throw. The corre- sponding cross ratios R (MN, A A') and B (NM, AA*) are reciprocals of each other (cf. Theorem 13, Cor. 3, Chap. VI). 205 206 ONE-DIMENSIONAL PROJECTIVITIES [CHAP.VIII COROLLARY 1. A projectivity on a line with two given distinct double points is uniquely determined ly its characteristic throw or cross ratio. COROLLARY 2. The characteristic cross ratio of any involution with double points is 1. This follows directly from Theorem 27, Cor. 1, Chap. IV, and Theorem 13, Cor. 2, Chap. VI. If m, n are nonhomogeneous coordinates of the double points, and Jc is the characteristic cross ratio of a projectivity on a line, we have x t m f x m_ , x 1 n ' x n for every pair of homologous points x, x f . This is the analytic expres- sion of the above theorem, and leads at once to the following analytic expression for a projectivity on a line with two distinct double points m, n : COROLLARY 3. Any projectivity on a line with two distinct double points m, n may be represented ~by the equation x'n x n x r , x being any pair of homologous points. For when cleared of fractions this is a bilinear equation in x j , x which obviously has m, n as roots. Moreover, since any projectivity with two given distinct double points is uniquely determined by one additional pair of homologous elements, it follows that any projec- tivity of the kind described can be so represented, in view of the fact that one such pair of homologous points will always determine the multiplier k These considerations offer an analytic proof of Theo- rem 1, for the case when the double points Jf, 2f are distinct. It is to be noted, however, that the proof of Theorem 1 applies equally well when the points M, N coincide, and leads to the follow- ing theorem : THEOREM 2. If in a paralolic projectivity with double point M the points A A 1 and BB f are two pairs of homologous points , the parabolic projectivity with double point M which puts A into B also puts A 1 into B'. COROLLARY. The characteristic cross ratio of any parabolic projcc- timty is unity. 73] CHARACTERISTIC THROW 207 The characteristic cross ratio together with the double point is therefore not sufficient to characterize a parabolic projectivity com- pletely. Also, the analytic form for a projectivity with double points m, n, obtained above, breaks down when m = n. We may, however, readily derive a characteristic property of parabolic projectivities, from which will follow an analytic form for these projectivities. THEOREM 3. If a parabolic projectivity with doiible point M trans- forms a point A into A! and A f into A fl , the pair of points A, A rf is harmonic with the pair A f M; i.e. we have H (MA r , AA ff ). Proof. By Theorem 23, Chap. IV, we have Q(MAJL*, MA" A'). Analytically, if the coordinates of M 9 A, A r , A rf are m, %, x r , x n respectively, we have, by Theorem 13, Cor. 4, Chap. VI, 2 = 1 1 x ffit x " in x - yn This gives x'm x m x n m x'm which shows that if each member of this equation be placed equal to t, the relation (1) ' m x is satisfied by every pair of homologous points of the sequence obtained by applying the projectivity successively to the points A, A r , A", It is, however, readily seen that this relation is satisfied by every pair of homologous points on the line. Tor relation (1), when cleared of fractions, clearly gives a bilinear form in x f and x, and is therefore a projectivity; and this projectivity clearly has only the one double point m. It therefore represents a parabolic projectivity with the double point m y and must represent the projectivity in question, since the relation is satisfied by the coordinates of the pair of homologous points A, A\ which are sufficient with the double point to determine the projectivity. We have then : COROLLABY 1. Any parabolic projectivity with a double point, M, may be represented by the relation (1). DEFINITION. The number t is called the characteristic constant of the projectivity (1). 208 ONE-DIMENSIONAL PEOJECTIVITIES [CHAP. VIII COROLLARY 2. Conversely, if a projectivity with a double point M transforms a point A into A', and A* into A !r , such that we have H (MA 1 , -4-4"), the projectwity is parabolic. Proof. The double point M and the two pairs of homologous points AA r , A'A n are sufficient to determine the projectivity uniquely; and there is a parabolic projectivity satisfying the given conditions. 74. Projective projectivities. Let TT be a projectivity on a line I, and let TT I be a projectivity transforming the points of I into the points of another or the same line V. The projectivity Ti^Tnrf 1 is then a projectivity on V. For Tr^ 1 transforms any point of V into a point of Z, TT transforms this point into another point of I, which in turn is transformed into a point of V by ir v Thus, to every point of V is made to correspond a unique point of l f , and this correspondence is projec- tive, since it is the product of projective correspondences. Clearly, also, the projectivity ir^ transforms any pair of homologous points of TT into a pair of homologous points of w^w-Trf 1 . DEFINITION. The projectivity Tj^Trrrf 1 is called the transform of V "by TT X ; two projectivities are said to be projectile or conjugate if one is a transform of the other by a projectivity. The question now arises as to the conditions under which two pro- jectivities are projective or conjugate. A necessary condition is evi- dent. If one of two conjugate projectivities has two distinct double points, the other must likewise have two distinct double points; if one has no double points, the other likewise can have no double points ; and if one is parabolic, the other must be parabolic. The further conditions are readily derivable in the case of two projectivities with distinct double points and in the case of two parabolic projectivities. They are stated in the two following theorems : THEOREM 4. Two projectivities each of which has two distinct double points are conjugate if and only if their characteristic throws are equal. Proof. The condition is necessary. For if TT, TT' are two conjugate projectivities, any projectivity T^ transforming TT into TT' transforms the double points M, N of TT into the double points M', N f of TT', and also transforms any pair of homologous points A, A l of TT into a pair of homologous points A 1 , A[ of ir 1 ; i.e. Tr^MNAA^ = M'N'A'Af. But this states that their characteristic throws are equal. 74, 75] GKOUPS ON A LINE 209 The condition is also sufficient ; for if it is satisfied, the projec- tivity TT^ defined by clearly transforms TT into TT'. COROLLARY. Any two involutions with doiible points. are conjugate. THEOREM 5. Any two parabolic projectivities are conjugate. Proof. Let the two parabolic projectivities be defined by ir(MMA) = MMA V and 7r r (M r M'A r ) = Jf'Jf^'. Then the projectivity ^ defined by ir^MAAJ = M'A'Af clearly transforms TT into TT'. Since the characteristic cross ratio of any parabolic projectivity is unity, the condition of Theorem 4 may also be regarded as holding for parabolic projectivities. 75. Groups of projectivities on a line. DEFINITION. Two groups G and G ; of projectivities on a line are said to be conjugate if there exists a projectivity ir^ which transforms every projectivity of G into a projectivity of G', and conversely. We may then write 7r 1 G'7r~ ;l =G / ; and G' is said to be the transform of G ly nr. We have already seen (Theorem 8, Chap. Ill) that the set of all projectivities on a line form a group, which is called the general pro- jective group on the line. The following are important subgroups : 1. TJie set of all projectivities leaving a given point of the line invariant. Any two groups of this type are conjugate. For any projectivity transforming the invariant point of one group into the invariant point of the other clearly transforms every projectivity of the one into some projectivity of the other. Analytically, if we choose % = oo as the invariant point of the group, the group consists of all projectivities of the form x r = ase + 'b. 2. The set of all projectivities leaving two given distinct points invariant * Any two groups of this type are conjugate. Tor any projectivity transforming the two invariant points of the one into the invariant points of the other clearly transforms every projectivity of the one 210 ONE-DIMENSIONAL PEOJECTIYITIES [CHAP.VIII into a projectivity of tlie other. Analytically, if x v x 2 are the two invariant points, the group consists of all projectivities of the form AU( ____ x f ~ The product of two such projectivities with multipliers k and Id is clearly given by x f x z This shows that any two projectivities of this group are commuta- tive. This result gives THEOREM 6. Any two projectivities which have two double points in common are commutative. This theorem is equivalent to the commutative law for multiplication. If the double points are the points and oo, the group consists of all projec- tivities of the form x f = ax. 3. The set of all parabolic projectivities with a common double point In order to show that this set of projectivities is a group, it is only necessary to show that the product of two parabolic projectivities with the same double point is parabolic. This follows readily from the analytic representation. The set of projectivities above described consists of all transformations of the form ^ l where x^ is the common double point (Theorem 3, Cor. 1). If ___ /VM .. fy> f)i __ /y ifj " " IAJ, tfj ^^ tAj* are two projectivities of this set, the product of the first by the second is given by j j -. =: ' ~r & T" ^o, a/ x l x %i x which is clearly a projectivity of the set. It shows, moreover, that any two projectivities of this group are commutative. Whence THEOREM 7. Any two parabolic projectivities on a line with the same double point are commutative. This theorem is independent of Assumption P, although this assumption* is implied in the proof we have given. The theorem has already been proved without this assumption in Example 2, p. 70. 75] GBOUPS ON A LINE 211 Any two groups of this type are conjugate. For every projectivity transforming the double point of one group into the double point of the other transforms the one group into the other, since the projec- tive transform of a parabolic projectivity is parabolic. DEFINITION. Two subgroups of a group G are said to be conjugate under G if there exists a transformation of G which transforms one of the subgroups into the other. A subgroup of G is said to be self- conjugate or invariant under G if it is transformed into itself by every transformation of G; i.e. if every transformation in G trans- forms any transformation of the subgroup into another (or the same) transformation of the subgroup. We have seen that any two groups of any one of the three types are conjugate subgroups of the general projective group on the line. We may now give an example of a self-conjugate subgroup. Tlie set of all parabolic projectivities in a group of Type 1 alove is a self-conjugate subgroup of this group. It is clearly a subgroup, since it is a group of Type 3. And it is self-conjugate, since any conjugate of a parabolic projectivity is parabolic, and since every projectivity of the group leaves the common double point invariant. EXERCISES 1. Write the eqtiations of all the projective transformations which permute among themselves (a) the points (0, 1), (1,0), (1, 1) ; (&) the points (0, 1), (1,0), (1,1), (a,&); (c) the points (0,1), (1,0), (1,1), (-1,1). What are the equations of the self-conjugate subgroup of the group of transforma- tions (a)? 2. If a projectivity x r (ax + I}/ (ex -f d) having two distinct double ele- ments be written in the form of Cor. 3, Theorem 1, show that and that ad be 3. If a parabolic projectivity x f = (ax + b)/(cx + d) be written in the form of Theorem 3, Cor. 1, show that x^ = (a d)/2 c, and t = 2 c/(a + ?). 4. Show that a projectivity with distinct double points x v x z and charac- teristic cross ratio k can be written in the form 1 or 1 a?! 1 1 ar k 1 212 ONE-DIMENSIONAL PEOJECTIVITIES [CHAP.VIII 5. Show that the parabolic projectivity of Theorem 3, Cor. 1, may be written in the form x i ^i 1 1 1 x 1 1 i 1 1 6. If by means of a suitably chosen transformation of a group any of the elements transformed may be transformed into any other element, the group is said to be transitive. If by a suitably chosen transformation of a group any set of n distinct elements may be transformed into any other set of n distinct elements, and if this is not true for all sets of n + 1 distinct elements, the group is said to be n-ply transitive. Show that the general projective group on a line is triply transitive, and that of the subgroups listed in 75 the first is doubly transitive and the other two are simply transitive. 7. If two projecti vities on a line, each having two distinct double points, have one double point in common, the characteristic cross ratio of their prod- uct is equal to the product of their characteristic cross ratios. 76. Projective transformations between conies. "We have consid- ered hitherto projectivities between one-dimensional forms of the first degree only. We shall now see how projectivities exist also be- tween one-dimensional forms of the second degree, and also between a one-dimensional form of the first and one of the second degree. Many familiar theorems will hereby appear in a new light. As typical for the one-dimensional forms of the second degree we choose the conic. The corresponding theorems for the cone then follow by the principle of duality. Let TTj be a projective collineation between two planes a, a v and let C z be any conic in a. Any two projective pencils of lines in a are then transformed by ^ into two projective pencils of lines in a v such that any two homologous lines of the pencils in a are trans- formed into a pair of homologous lines in o^; for if TT be the projec- tivity between the pencils in a', T^TTTrf 1 will be a projectivity between the pencils in a (cf. 74). Two projective pencils of lines generating the conic <7 2 thus correspond to two pencils of lines in a generating a conic Cf. The transformation ir^ then transforms every point of C 2 into a unique point of Cf. Similarly, it is seen that ir^ transforms every tangent of (7 2 into a unique tangent of Cf. DEFINITION. Two conies are said to be protective if to every point of one corresponds a poinjb of the other, and to every tangent of one 76] TEAJNTSFOEMATION OF COJSTICS 213 corresponds a tangent of the other, in such a way that this correspond- ence may be brought about by a projective collineation between the planes of the conies. The projective collineation is then said to generate the projectivity between the conies. Two conies in different planes are projective, for example, if one is the pro- jection of the other from a point on neither of the two planes. If the second of these is projected back on the plane of the first from a new center, we obtain two conies in the same plane that are projective. We will see presently that two projective conies may also coincide, in which case we obtain a pro- jectivity on a conic. THEOREM 8. Two conies that are projectile with a third are projective. Proof. This is an immediate consequence of the definition and the fact that the resultant of two collineations is a collineation. We proceed now to prove the fundamental theorem for projec- tivities between two conies. THEOREM 9. 'A projectivity "between two conies is uniquely deter- mined if three distinct points (or tangents) of one are made to corre- spond to three distinct points (or tangents] of the other. FIG. 86 Proof. Let <7 2 , <7 2 be the two conies (fig. 86), and let A, B, be three points of (7 2 , and A f , B r , C r the corresponding points of Cf. Let P and P f be the poles of AB and A'Jff with respect to <7 2 and <7 X respectively. If now the collineation TT is defined by the relation 7r(ABCP)=zA f B r C'P f (Theorem 18, Chap. IV), it is clear that the conic (7 2 is transformed by TT into a conic through the points A', B f , C r 9 with tangents A'P* and B f P f . This conic is uniquely determined by these specifications, however, and is therefore identical with CJ. The collineation TT then transforms C 2 into Cf in such a way that the points A, B, C are transformed into A 1 , B f , C f respectively. Moreover, 214 ONE-DIMENSIONAL PEOJECTIYITIES [CHAP.VIII suppose 7r f were a second collineation transforming C 2 into C* in the way specified. Then 7r'"V would be a collineation leaving A, B, C, P invariant ; i.e. TT = TT'. The argument applies equally well if A f B'C f are on the conic O 2 , i.e. when the two conies 2 , C? coincide. In this case the projectivity is on the conic C. This gives COROLLARY 1. A projectivity on a conic is uniquely determined when three pairs of homologous elements (points or tangents] are given. Also from the proof of the theorem follows COROLLARY 2. A collineation in a plane which transforms three distinct points of a conic into three distinct points of the same conic and which transforms the pole of the line joining two of the first three points into the pole of the line joining the two corresponding points transforms the conic into itself. The two following theorems establish the connection between pro- jectivities between two conies and projectivities between one-dimen- sional forms of the first degree. THEOREM 10. If A and B 1 are THEOREM 10'. If a and V are any two points of two projective any two tangents of two protective conies C* and Of respectively, the conies C z and C* respectively, the pencils of lines with centers at A pencils of points on a and V are and B ! are protective if every pair protective if every pair of homol- of homologous lines of these pencils ogous points on these lines is on pass through a pair of homologous a pair of homologous tangents of points on the two conies respectively, the conies respectively. Proof. It will suffice to prove the theorem on the left. Let A' be the point of (7f homologous with A. The collineation which generates the projectivity between the conies then makes the pencils of lines at A and A f protective, in such a way that every pair of homologous lines contains a pair of homologous points of the two conies. The pen- cil of lines at B f is projective with that at A 1 if they correspond in such a way that pairs of homologous lines intersect on C* (Theorem 2, Chap. V). This establishes a projective correspondence between the pencils at A and B r in which any two homologous lines pass through two homologous points of the conies and proves the theorem. It should be noted that in this projectivity the tangent to C 2 at A corresponds to the line of the pencil at B' passing through A f . 70] TBAJtfSFOKMATION OF COLICS 215 COROLLARY. Conversely, if two conies correspond in such a way that every pair of homologous tan- gents is on a pair of homologous points of two projective pencils of points whose axes are tangents of the conies, they are protective. COROLLARY. Conversely, if two conies correspond in such a way that every pair of homologoits points is on a pair of homologous lines of two protective pencils of lines whose centers are on the conics 9 they are projective. Proof. This follows from the fact that the projectivity between the pencils of lines is uniquely determined by three pairs of homologous lines. A projectivity between the conies is also determined by the three pairs of points (Theorem 9), in which three pairs of homolo- gous 4 lines of the pencils meet the conies. But by what precedes and the theorem above, this projectivity is the same as that described in the corollary on the left. The corollary on the right may be proved similarly. If the two conies are in the same plane, it is simply the plane dual of the one on the left. By means of these two theorems the construction of a projectivity between two conies is reduced to the construction of a projectivity between two primitive one-dimensional forms. It is now in the spirit of our previous definitions to adopt the following : DEFINITION. A Hue conic and a pencil of points whose axis is a line of the conic are said to be perspective if they correspond in such a way that every line of the conic passes through the homolo- gous point of the pencil of points. A line conic and a pencil of lines are said to be perspective if every two homologous lines meet in a point of a pencil of points whose DEFINITION. A point conic and a pencil of lines whose center is a point of the conic are said to be perspective if they correspond in such a way that every point of the conic is on the homologous line of the pencil. A point conic and a pencil of points are said to be perspective if every two homol- ogous points are on the same line of a pencil of lines whose center is a point of the conic. axis is a line of the conic. The reader will now readily verify that with this extended use of the term perspective, any sequence of perspectivities leads to a pro- jectivity. For example, two pencils of lines perspective with the same point conic are projective by Theorem 2, Chap. V; two point conies 216 ONE-DIMENSIONAL PEOJECTIVITIES [CHAP.VIII perspective with the same pencil of lines or with the same pencil of points are projective by Theorem 10, Cor., etc. Another illustration of this extension of the notion of perspectivity leads readily to the following important theorem : THEOEEM 11. Two conies which are not in the same plane and have a common tangent at a point A are sections of one and the same cone. Proof. If the two conies (7 2 , C* (fig. 87) are made to correspond in such a way that every tangent x of one is associated with that tangent #'of the other which meets x in a point of the common tangent a of the conies, they are projective. For the tangents of the conies are then 87 perspective with the same pencil of points (cf. Theorem 10', Cor.). Every pair of homologous tangents of the two conies determines a plane. If we consider the point of intersection of three of these planes, say, those determined by the pairs of tangents W, erf, dd f , and project the conic C* on the plane of <7 2 from 0, there results a conic in the plane of C*. This conic has the lines 6, c, d for tangents and is tangent to a at A ; it therefore coincides with C 2 (Theorem 6', Chap. V). The two conies (7 2 , C? then have the same projection from 0, which proves the theorem.* EXERCISES 1. State the theorems concerning cones dual to the theorems of the preced- ing sections. 2. By dualizing the definitions of the last article, define what is meant by the perspectivity between cones and the primitive one-dimensional forms. 3. If two projective conies have three self-corresponding points, they are perspective with a common pencil of lines. 4. If two projective conies have four self -corresponding elements, they coincide. 5. State the space duals of the last two propositions. *It will be seen later that this theorem leads to the proposition that any conic may be obtained as the projection of a circle tangent to it in a different plane. 76, 77] PEOJECTIVITIES OX A CONIC 217 6. If a pencil of lines and a conic in the plane of the pencil are projective, but not perspective, not more than three lines of the pencil pass through their homologous points on the conic. (Hint. Consider the points of intersection of the given conic with the conic generated by the given pencil and a pencil of lines perspective with the given conic.) Dualize. 7. The homologous lines of a line conic and a projective pencil of lines in the same plane intersect in points of a " curve of the third order " such that any line of the plane has at most three points in. common with it. (This fol- lows readily from the last exercise.) 8. The homologous elements of a cone of lines and a projective pencil of planes meet in a " space curve of the third order" such that any plane has at most three points in common with it. 9. Dualize the last two propositions. 77. Projectivities on a conic. We have seen that two protective' conies may coincide (Theorems 8-10), in which case we obtain a projective correspondence among the points or the tangents of the conic. The construction of the projectivity in this case is very simple, and leads to many important results. It results from the following theorems": THEOREM 12. If A, A f are any THEOREM 12'. If a, a f are any two distinct homologous points of two distinct homologous tangents a projectivity on a conic, and B,3 ! ; of a projectivity on a conic, and C, C r ; etc., are any other pairs of I, V; c, c f ; etc., are any other pairs 218 ONE-DIMENSIONAL PEOJECTIVITIES [CHAP, vin homologous points, the lines A r B of homologous tangents, the points and AB', A f C and AC 1 , etc., meet a f b and aV, a f c and ac f , etc., are in points of the same line; and collinear with the same point; this line is independent of the pair and this point is independent of AAl chosen. the pair aa 1 chosen. Proof. The pencils of lines A* (ABC- - ) and A(A'B f C f < - ) are pro- jective (Theorem 10), and since they have a self-corresponding line AA 1 , they are perspective, and the pairs of homologous lines of these two pencils therefore meet in the points of a line (fig. 88). This proves the first part of the theorem on the left. That the line thus determined is independent of the homologous pair AA! chosen then follows at once from the fact this line is the Pascal line of the simple hexagon AB'CA'BC 1 , so that the lines B J G and BC f and all other analogously formed pairs of lines meet on it. The theorem on the right follows by duality. DEFINITION. The line and the point determined by the above dual theo- rems are called the axis and the center of the projectivity respectively. COROLLARY 1. A (nonidentical) COROLLARY 1'. A (nonidenti- projectivity on a conic is wiiquely cal) projectivity on a conic is determined when the axis of ho- wniquely determined when the mology and one pair of distinct center and one pair of distinct homologous points are given. homologous tangents are given. These corollaries follow directly from the construction of the pro- jectivity arising from the above theorem. This construction is as follows : Given the axis o and a pair of distinct homologous points AA f , to get the point P 1 homologous with any point P on the conic ; join P to A! ; the point P f is then on the line joining A 'to the point of intersection of A'P with o. Or, given the center and a pair of distinct homologous tangents aa f , to construct the tangent p f homolo- gous with any tangent p ; the line joining the point a r p to the center meets a in a point of p r . COROLLARY 2. Every double COROLLARY 2'. Every doulle point of a projectivity on a conic line of a projectivity on a conic is on the axis of the projectivity ; contains the center of the projec- and, conversely, every point com- tivity; and, conversely, every tan- mon to the axis and the conic is gent of a conic through the center a double point. is a double line of the projectivity. 77] PEOJECTIYITIES ON A CONIC 219 COROLLARY 3. A projectimty COROLLARY 3'. A projectimty among the points on a conic is among the tangents to a conic is parabolic if and only if the axis parabolic if and only if the center is tangent to the conic. . is a point of the conic. THEOREM 13. A projectimty among the points of a conic determines a projectimty of the tangents in which the tangents at pairs of homol- ogous points are homologous. Proof. This follows at once from the fact that the collineation in the plane of the conic which generates the projectivity transforms the tangent at any point of the conic into the tangent at the homol- ogous point, and hence also generates a projectivity between the tangents. THEOREM 14. The center of a projectivity of tangents on a conic and the axis of the corresponding projectimty of points are pole and polar with respect to the conic. A FIG. 89 Proof. Let AA ! , BB ! t CC ! (fig. 89) he three pairs of homologous points (AA 1 being distinct), and let A'B and AB ! , A'C and AC f , meet in points E and S respectively, which determine the axis of the pro- jectivity of points. Now the polar of E with respect to the conic is determined by the intersections of the pairs of tangents at A f , B and A) B 1 respectively; and the polar of S is ' determined by the pairs of tangents at A 1 , C and A, C r respectively (Theorem 13, Chap. V). The pole of the axis JZS is then determined as the intersection of these 220 ' ONE-DIMENSIONAL PEOJEOTIVITIES [CHAP.VIII two polars (Theorem 17, Chap. V), But by definition these two polars also determine the center of the projectivity of tangents. This theorem is obvious if the projectivity has double elements ; the proof given, however, applies to all cases. The collineation generating the projectivity on the conic transforms the conic into itself and clearly leaves the center and axis invariant. The set of all collineations in the plane leaving the conic invariant form a group (cf. p. 67). In determining a transformation of this group, any point or any line of the plane may be chosen arbitrarily as a double point or a double line of the collineation ; and any two points or lines of the conic may be chosen as a homologous pair of the collineation. The collineation is then, however, uniquely deter- mined. In fact, we have already seen that the projectivity on the conic is uniquely determined by its center and axis and one pair of homologous elements (Theorem 12, Cor. 1); and the theorem just proved shows that if the center of the projectivity is given, the axis is uniquely determined, and conversely. COROLLARY 1. A plane protective collineation which leaves a non- degenerate conic in its plane invariant is of Type I if it lias two doutle points on the conic, tmless it is of period tivo, in which case it is of Type IV; and is of Type III if the corresponding projectivity on the conic is parabolic. COROLLARY 2. An elation or a collineation of Type II transforms every nondegenerate conic of its plane into a different conic. COROLLARY 3. A plane $rojective collineation which leaves a conic in its plane invariant and has no double point on the conic has one and only one double point in the plane. THEOREM 15. The group of protective collineations in a plane leav- ing a nondegenerate conic invariant is simply isomorphic* with the general protective group on a line. Proof. Let A be any point of the invariant conic. Any projectivity on the conic then gives rise to a projectivity in the flat pencil at A in which two lines are homologous if they meet the conic in a pair of homologous points. And, conversely, any projectivity in the flat * Two groups are said to be simply isomorpMc if it is possible to establish a (1,1) correspondence between the elements of the two groups such that to the product of any two elements of one of the groups corresponds the product of the two corre- sponding elements of the other. 77,78] INVOLUTIONS 221 pencil at A gives rise to a projectivity on the conic. The group of all projectivities on a conic is therefore simply isomorphic with the group of all projectivities in a fiat pencil, since it is clear that in the corre- spondence described between the projectivities in the flat pencil and on the conic, the products of corresponding pairs of projectivities will be corresponding projectivities. Hence the group of plane collineations leaving the conic invariant is simply isomorphic with the general pro- jective group in a flat pencil and hence with the general projective group on a line. 78. Involutions. An involution was defined (p. 102) as any projec- tivity in a one-dimensional form which is of period two, ie. by the relation I 2 = 1 (I = 1), where I represents an involution. This relation is clearly equivalent to the other, I = I~ 1 (l= 1), so that any projec- tivity (not the identity) in a one-dimensional form, which is identical with its inverse, is an involution. It will be recalled that since an in- volution makes every pair of homologous elements correspond doubly, i.e. A to A! and A f to A, an involution may also be considered as a pairing of the elements of a one-dimensional form ; any such pair is then called a conjugate, pair of the involution. We propose now to consider this important class of projectivities more in detail. To this end it seems desirable to collect the fundamental properties of invo- lutions which have been obtained in previous chapters. They are as follows : 1. If the relation 7r z (A) A Jwlds for a single element A (not a double element of TT) of a one-dimensional form, the projectivity TT is an involution, and the relation holds for every element of the form (Theorem 26, Chap. IV). 2. An involution is uniquely determined when two pairs of conjib- gate elements are given (Theorem 26, Cor., Chap. IV). 3. The opposite pairs of any quadrangular set are three pairs of an involution (Theorem 27, Chap. IV). 4. If M, N are distinct double elements of any projectivity in a one-dimensional form and A, A r and B, B l are any two pairs of homologous elements of the projectivity, the pairs of elements MN 9 AB J A f B are three pairs of an involution (Theorem 27, Cor. 3, Chap. IV), 5. If M, N are double elements of an involution, they are distinct, and every conjugate pair of the involution is harmonic with M, N (Theorem 27, Cor. 1, Chap. IV). 222 ONE-DIMENSIONAL PEOJECTIVITIES [CHAP, vm 6. An involution is uniquely determined, if two double elements are given, or if one double element and another conjugate pair are given. (This follows directly from the preceding.) 7. An involution is represented analytically ly a bilinear form cscx r a (JG + x') & = 0, or ly the transformation ex a (Theorem 12, Cor. 3, Chap. YI). 8. An involution with double elements m, n may be represented ly the transformation x' (Theorem 1, (Tors. 2, 3, Chap. VIII). We recall, finally, the Second Theorem of Desargues and its various modifications ( 46, Chap. V), which need not be repeated at this place. It has been seen in the preceding sections that any projec- tivity in a one-dimensional primitive form may he transformed into a projectivity on a conic. We shall find that the construction of an in- volution on a conic is especially simple, and may be used to advantage in deriving further properties of involutions. Under duality we may confine our consideration *\ to the case of an involu- ' tion of points on a conic. y THEOREM 16. The lines joining the conjugate points of an involution on a conic all pass through the center of the involution. Proof. Let A, A' (fig. 90) be any conjugate pair (A not a double point) of an involution of points on a conic C 2 . The line A A! is then an invariant line of the collineation gener- ating the involution. Every line joining a pair of distinct conjugate points of the involution is therefore invariant, and the generating collineation must be a perspective collineation, since any collineation leaving four lines invariant is either perspective or the identity . 90 78] INVOLUTIONS 223 (Theorem 9, Cor. 3, Chap. III). It remains only to show that the center of this perspective collineation is the center of the involution. Let B y B f (B not a double point) be any other conjugate pair of the involution, distinct from A, A 1 . Then the lines AB f and A'B inter- sect on the axis of the involution. But since B t B r correspond to each other doubly, it follows that the lines AB and A f B f also intersect on the axis. This axis then joins two of the diagonal points of the quadrangle AA ! BB r . The center of the perspective collineation is determined as the intersection of the lines AA r and BB ! , i.e. it is the third diagonal point of the quadrangle AA'BB*. The center of the collineation is therefore the pole of the axis of the involution (Theorem 14, Chap. V) and is therefore (Theorem 14, above) the center of the involution. Since this center of the involution is clearly not on the conic, the generating collineation of any involution of the conic is a homology, whose center and axis o are pole and polar with respect to the conic. A homology of period two is sometimes called a harmonic homol- ogy, since it transforms any point P of the plane into its harmonic conjugate with respect to and the point in which OP meets the axis. It is also called a protective reflection or a point-line reflec- tion. Clearly this is the only kind of homology that can leave a conic invariant. The construction of the pairs of an involution on a conic is now very simple. If two conjugate pairs A 9 A* and B, B f are given, the lines AA 1 and BB J determine the center of the involution. The conjugate of any other point C on the conic is then determined as the intersec- tion with the conic of the line joining C to the center. If the involu- tion has double points, the tangents at these points pass through the center of the involution ; and, conversely, if tangents can be drawn to the conic from the center of the involution, the points of contact of these tangents are double points of the involution. The great importance of involutions is in part due to the following theorem : THEOREM 17. Any projectimty in a one-dimensional form may le obtained, as the product of two involutions. Proof. Let II be the projectivity in question, and let A be any point of the one-dimensional form which is not a double point. 224 ONE-DIMENSIONAL PEOJECTIVITIES [CHAP.VIII Further, let U(A) = A! and ft (A 1 ) = A". Then, if I, is the involution of which A 1 is a double point and of which AA n is a conjugate pair (Prop. 6, p. 222), we have so that in the projectivity I^II the pair AA f corresponds to itself doubly. I^II is therefore an involution (Prop. 1, p. 221). If it be denoted by I 2 , we have 1^11 = 1^ or TL^I^I^ which was to be proved. This proof gives at once : COROLLARY 1. Any projectivity II may le represented as the prod- uct of two involutions, II = I 1 -I 2 , either of which (but not lotli) has an arbitrary point (not a double point of II) for a double point. Proof. We have seen above that the involution ^ may have an arbitrary point (A 1 ) for a double point. If in the above argument we let I 2 be the involution of which A f is a double point and AA rf is a conjugate pair, we have II I & (A'A' r ) = A r! A r ; whence II I 2 is an invo- lution, say I r We then have II = 1^ I 2 , in which I 2 has the arbitrary point A f for a double point. The argument given above for the proof of the theorem applies without change when A = A ff , i.e. when the projectivity II is an in- volution. This leads readily to the following important theorem : COROLLARY 2. If A A! is a conjugate pair of an involution I, the involution of which A, A! are double points transforms I into itself, and the two involutions are commutative. Proof. The proof of Theorem 17 gives at once I I^Lj, where \ is determined as the involution of which A,A f are double points. We have then I x - 1 = I 2 , from which follows, by taking the inverse of both sides of the equality, I ^ = I^ 1 = I 2 , or l l - 1 = ! I a , or I x I I x = I. As an immediate corollary of the preceding we have COROLLARY 3. The product of two involutions with double points A, A f and B, B ! respectively transforms into itself the involution in which A A f and B B' are two conjugate pairs. Involutions related as are the two in Cor. 2 above are worthy of special attention. DEFINITION. Two involutions are said to be harmonic if their product is an involution. 78,79] INVOLUTIONS 225 THEOREM 18. Two harmonic involutions are commutative. Proof. If I v I 2 are harmonic, we have, by definition, I x - 1 2 = I 3 , where I 3 is an involution. This gives at once the relations 1^ I 2 - 1 3 = 1 and COROLLARY. Conversely, if two distinct involutions are commutative, they are harmonic. For from the relation Ij-I^Ig-^ follows (I 1 I 2 ) 2 = 1; i.e. I^I^ is an involution, since I^I^l. DEFINITION. The set of involutions harmonic with a given involu- tion is called a pencil of involutions. It follows then from Theorem 17, Cor. 2, that the set of all involu- tions in which two given elements form a conjugate pair is a pencil. Thus the double points of the involutions of such a pencil are the pairs of an involution. 79. Involutions associated with a given projectivity* In deriving further theorems on involutions we shall find it desirable to suppose the projectivities in question to be on a conic. THEOREM 19. If a projectivity on a conic is represented as the product of two involutions, the axis of the projectivity is the line joining the centers of the two involutions. Proof. Let the given projec- tivity be II = I 2 - I x ; I v I 2 being two involutions. Let O v 2 be the centers of I x , I 2 respectively (fig. 91), and let "A and B be any two points on the conic which are not double points of either of the involutions \ or I 2 and which are not a conjugate pair of I t or I 3 . If, then, we have II (AB) = A'B 1 , we have, by hypothesis, \(AB) = A^B^ and ) = A f B'- ? AV B 1 being uniquely determined points of the conic, B such that the lines AA V BB^ intersect in 1 and the lines A^AI 9 intersect in 2 . The Pascal line of the hexagon AA^BB^ff then passes through O v O z and the intersection of the lines AB 1 and A'B. But the latter point is a point on the axis of II. This proves the theorem. 226 ONE-DIMENSIONAL PEOJECTIVITIES [CHAP, vni COROLLARY. A projectivity on a conic is the product of two involu- tions, the center of one of which may le any arbitrary point (not a double point) on the axis of the projectivity ; the center of the other is then uniquely determined. Proof. Let the projectivity II be determined by its axis I and any pair of homologous points A, A 1 (fig. 91). Let 1 be any point on the axis not a double point of II, and let I x be the involution of which O x is the center. If, then, l l (A) = A v the center 2 of the involution I such that II = I 3 - I x is clearly determined as the intersection of the line A^A! with the axis. For by the theorem the product / 2 J a is a projectivity having I for an axis, and it has the points A, A f as a homol- ogous pair. This shows that the center of the first involution may be any point on the axis (not a double point). The modification of this argument in order to show that the center of the second involu- tion may be chosen arbitrarily (instead of the center of the first) is obvious. THEOREM 20. There is one and only one involution commutative with a given nonparabolic noninvolutoric projectivity. If the projec- tivity is represented on a conic, the center of this involution is the center of the projectivity. Proof. Let the given nonparabolic projectivity II be on a conic, and let I be any involution commutative with H ; Le. such that we have II I = I- II. This is equivalent to II I II"" 1 = I. That is to say, I is transformed into itself by II. Hence the center of I is transformed into itself by the collineation generating II. But by hypothesis the only invariant points of this collineation are its center and the points (if existent) in which its axis meets the conic. Since the center of I cannot be on the conic, it must coincide with the center of II. More- over, if the center of I is the same as the center of II, I is trans- formed into itself by the collineation generating II, II -LIT" * = I. Hence H I = I H. Hence I is the one and only involution commu- tative with II, COROLLARY 1. There is no involution commutative with a parabolic projectivity. DEFINITION. The involution commutative with a given nonpara- bolic noninvolutoric projectivity is called the involution belonging to the given projectivity. An involution belongs to itself. 79] INVOLUTION'S 227 COROLLARY 2. If a nonparabolic projectivity has double points, the involution belonging to the projectivity has the same double points. For if the axis of the projectivity meets the conic in two points, the tangents to the conic at these points meet in the pole of the axis. It is to be noted that the involution I belonging to a given projec- tivity II transforms II into itself, and is transformed into itself by II. Indeed, from the relation II I = I II follow at once the relations I.II-I = II and DM'II~' 1 = L Conversely, from the equation iM-n- 1 follows n.i = Mi. THEOREM 21. The necessary and sufficient condition that two invo- lutions on a conic be harmonic is that their centers be conjugate with respect to the conic. Proof. The condi- 3 \^^-^ A tion is sufficient. For let I 1? I 2 be two invo- lutions on the conic whose centers O v 2 respectively are con- jugate with respect to the conic (fig. 92). Let A be any point of the conic not a double point of either involution, and let I 1 (A)^A 1 and I Z (AJ=A'. If, then, l l (A f ) = A[, the center O l is a diagonal point of the quadrangle AA^A!A[, and the center 2 is on the side A^A 1 . Since, by hypothesis, 2 is conjugate to O l with respect to the conic, it must be the diago- nal point on AjA f , i.e. it must be collinear with AA[. We have then l z >I^(AA J )A f A, i.e. the projectivity l^\ is an involution I 8 . The center 3 of the involution I 3 is then the pole of the line 1 3 with respect to the conic (Theorem 19). The triangle 1 a 8 is therefore self-polar with respect to the conic. It follows readily also that the condition is necessary. Tor the relation I^Ig Ig leads at once to the relation 1^ = 1^1,. If O v 3 , 3 are the centers respectively of the involutions I v I 2 , I 3 , the former of these two relations shows (Theorem 19) that B is the pole of the line 0^; while the latter shows that 3 is the pole of the line 0^0 y The triangle 0^0. 2 9 is therefore self-polar. 228 ONE-DIMENSIONAL PEOJECTIVITIES [CHAP, vm COROLLARY 1. Given any two involutions, there exists a third invo- lution which is harmonic with each of the given involutions. For if we take tlie two involutions on a conic, the involution whose center is the pole with respect to the conic of the line joining the centers of the given involutions clearly satisfies the condition of the theorem for each of the latter. COROLLARY 2. Three involutions each of which is harmonic to the other two constitute, together with the identity, a group. COROLLARY 3. The centers of all involutions in a pencil of involu- tions are collinear. THEOREM 22. The set of all projectivities to which belongs the same involution I forms a commutative group. Proof. If II, IIj are two projectivities to each of which belongs the involution I, we have the relations I -II -I = 13 and I-II 1 -I=:II 1 , from which follows I-II~ 1 I = n~ 1 and, by multiplication, the rela- , which shows that the set forms a group. To show that any two projectivities of this group are commu- tative, we need only sup- pose the projectivities given on a conic. Let A be any point on this conic, and let II (A) = A ! and TL l (A r ) = A, so that ^ 1 / . Since the Hani I.II.M.IVI=] A A, same involution I belongs, by hypothesis, both to II and 11^ these two projectivities have the same axis ; let it be the line I (fig. 93). The point IL^ (A) = A 1 is now readily determined (Theorem 12) as the intersection with the conic of the line joining A f to the intersection of the line AA[ with the axis I. In like manner, H(A^) is determined as tie intersection with the conic of the line joining A to the intersection of the line A^A r with the axis I Hence H (AJ = A[, and hence II U^A) = A[. It is noteworthy that when the common axis of the projectivities of this group meets the conic in two points, which are then common double points of all the projectivities of the group, the group is the 79] INVOLUTIONS 229 same as the one listed as Type 2, p. 209. If, however, our geometry admits of a line in the plane of a conic but not meeting the conic, the argument just given proves the existence of a commutative group none of the projectivities of which have a double point. THEOREM 23. Two involutions have a conjugate pair (or a double point) in common if and only if the product of the two invohctions has two double points (or is parabolic). Proof. This follows at once if the involutions are taken on a conic. For a common conjugate pair (or double point) must be on the line joining the centers of the two involutions. This line must then meet the conic in two points (or be tangent to it) in order that the involu- tions may have a conjugate pair (or a double point) in common. EXERCISES 1 . Dualize all the theorems and corollaries of the last two sections. 2. The product of two involutions on a conic is parabolic if and only if the line joining the centers of the involutions is tangent to the conic. Dualize. 3. Any involution of a pencil is uniquely determined when one of its con- jugate pairs is given. 4. Let II be a noninvolutoric projectivity, and let I be the involution be- longing to II; further, let TL(AA') = A' A", A being any point on which the projectivity operates which is not a double point, and let 1(4') = A. Show, by taking the projectivity on a conic, that the points A'A are harmonic with the points A A". 5. Derive the theorem of Ex. <4 directly as a corollary of Prop. 4, p. 221, assuming that the projectivity n has two distinct double points. 6. From the theorem of Ex. 4 show how to construct the involution be- longing to a projectivity n on a line without making use of any double points the projectivity may have. 7. A projectivity is uniquely determined if the involution belonging to it and one pair of homologous points are given. 8. The product of two involutions I 1? I 2 is a projectivity to which belongs the involution which is harmonic with each of the involutions I x , I 2 . 9. Conversely, every projectivity to which a given involution I belongs can be obtained as the product of two involutions harmonic with I. 10. Show that any two projectivities n i9 II 3 may be obtained as the product of involutions in the form H l = I'I 19 II 3 = I 3 -I; and hence that the product of the two projectivities is given by IVI^ = I 2 -I r 11. Show that a projectivity H = I-Ii may also be written IE = I 2 -I, I 2 being a uniquely determined involution ; and that in this case the two invo- lutions Ij, I 2 are distinct unless n is involutoric. 230 ONE-DIMENSIONAL PEOJECTIVITIES [CHAP.VIII 12. Show that if I x , I 2 , I 3 are three involutions of the same pencil, the relation (I^^-Ig) 2 ^ 1 must hold. 13. If aa', W, cc' are the coordinates of three pairs of points in involution, u J.T. 4. '- & &'- c c'-a - show that r = 1. a 7 c 6 a c' b 80. Harmonic transformations. The definition of harmonic involu- tions in the section above is a special case of a more general notion which can he defined for (1, 1) transformations of any kind whatever. DEFINITION. Two distinct transformations A and B are said to be harmonic if they satisfy the relation (AB" 1 ) 2 = 1 or the equivalent relation (BA- 1 )** 1, provided that AB" 1 ^ 1. A number of theorems which are easy consequences of this defini- tion when taken in conjunction with the two preceding sections are stated in the following exercises. (Of. C. Segre, Note sur les homo- graphies binaires et leur faisceaux, Journal fur die reine und ange- wandte Mathematik, Vol. 100 (1887), pp. 317-330, and H. Wiener, Ueber die aus zwei Spiegelungen zusammengesetzten Verwandt- schaften, Berichte d. K. sachsischen Gesellschaft der Wissenschaften, Leipag, Vol. 43 (1891), pp. 644-673.) EXERCISES 1. If A and B are two distinct involutoric transformations, they are har- monic to their product AB. 2. If three involutoric transformations A, B, T satisfy the relations (ABF) 2 = 1 , ABF 5* 1 , they are all three harmonic to the transformation AB. 3. If a transformation S is the product of two involutoric transformations A, B (i.e. 2 = AB) and T is an involutoric transformation harmonic to 21, then we have (ABr) 2 =l. 4. If A, J3, C, A', B', C' are six points of a line, the involutions A, B, T, such that T(AA') ~B'B, A(') = CC, *&(CC?)=A'A, are all harmonic to the same projectivity. Show that if the six points are taken on a conic, this proposition is equivalent to Pascal's theorem (Theorem 3, Chap. V). 5. The set of involutions of a one-dimensional form which are harmonic to a given nonparabolic projectivity form, a pencil. Hence, if an involution with double points is harmonic to a projectivity with two double points, the two pairs of double points form a harmonic set. 6. Let be a fixed point of a line I, and let C be called the mid-point of a pair of points A, J3 } provided that C is the harmonic conjugate of with respect to A and B. If A, B, C, A', B', C'are any six points of I distinct from 0, and AB' have the same mid-point as A'B, and BC' have the same mid-point as B'C> then CA' will have the same mid-point as C?A* 80,81] SCALE ON A CONIC 231 7. Any two involutions of the same one-dimensional form determine a pencil of involutions. Given two involutions A, B and a point M, show how to construct the other double point of that involution of the pencil of which one double point is M. 8. The involutions of conjugate points on a line I with regard to the conies of any pencil of conies in a plane with I form a pencil of involutions. 9. If two nonparabolic projectivities are commutative, the involutions belonging to them coincide, unless both projectivities are involutions, in which case the involutions may be harmonic. 10. If [II] is the set of projectivities to which belongs an involution I and A and B are two given points, then we have [II (.4)] -^ [II (By]. 11. A conic through two of the four common points of a pencil of conies of Type J meets the conies of the pencil in pairs of an involution. Extend this theorem to the other types of pencils of conies. Dualize. 12. The pairs of second points of intersection of the opposite sides of a complete quadrangle with a conic circumscribed to its diagonal triangle are in involution (Sturm, Die Lehre von den Geometrischen Verwandtschaften, Vol. I, p. U9). 81. Scale on a conic. The notions of a point algebra and a scale which we have developed hitherto only for the elements of one- dimensional primitive forms may also be studied to advantage on a conic. The constructions for the sum and the product of two points (numbers) on a conic is remarkably simple. As in the case on the line, let 0, 1, co be any three arbitrary distinct points on a conic C\ Regarding these as the fundamental points of our scale on the conic, the sum and the product of any two points x, y on the conic (which are distinct from co) are denned as follows : DEFINITION. The conjugate of in the involution on the conic having oo for a double point and x, y for a conjugate pair is called the sum of the two points x 9 y and is denoted by x + y (fig. 94, left). The conjugate of 1 in the involution determined on the conic by the conjugate pairs 0, oo and x, y is called the product of the points x, y and is denoted by x - y (fig. 94, right). It will be noted that under Assumption P this definition is entirely equivalent to the definitions of the sum and product of two points on a line, previously given (Chap. VI). To construct the point x + y on the conic (fig. 94), we need only determine the center of the involution in question as the intersection of the tangent at oo with the line joining the points x, y. The point x + y is then determined as the intersection with the conic of the line joining the center to the point 0. Similarly, 232 ONE-DIMENSIONAL PKOJECTIYITIES [CHAP. VIII to obtain the product of the points x, y we determine the center of the involution as the intersection of the lines Oo> and xy. The point x - y is then the intersection with the conic of the line joining this center to FIG. 94 the point 1. The inverse operations (subtraction and division) lead to equally simple constructions. Since the scale thus defined is obviously projective with the scale on a line, it is not necessary to derive again the fundamental properties of addition and subtraction, multiplication and division. It is clear from this consideration that the points of a conic form a field with reference to the operations just defined. This fact will be found of use in the analytic treatment of conies. At this point we will make use of it to discuss the existence of the square root of a number in the field of points. It is clear from the x FIG. 95 preceding discussion that if a number x satisfies the equation x* = a, the tangent to the conic at the point x must pass through the inter- section of the lines Ooo and 1 a (fig. 95). A number a will therefore have a square root in the field if and only if a tangent can le drawn to 81] SCALE ON A CONIC 233 the conic from the intersection of the lines Oco and 1 a; and, conversely, if the number a has a sqiiare root in the field, a tangent can be drawn to the conic from this point of intersection. It follows at once that if a number a has a square root x, it also has another which is obtained by drawing the second tangent to the conic from the point of inter- section of the lines Oco and la. Since this tangent meets the conic in a point which is the harmonic conjugate of x with respect to Oco, it follows that this second square root is x. It follows also from this construction that the point 1 has the two square roots 1 and 1 in any field in which 1 and 1 are distinct, i.e. whenever R Q is satisfied. We may use these considerations to derive the following theorem, which will be used later. THEOREM 2-4. If AA r , BB* are any two distinct pairs of an involu- tion, there exists one and only one pair CC f distinct from BB 1 such that the cross ratios %(AA r , BB>) B (AA r , CC r ) equal. Proof. Let the involution be taken on a conic, and let the pairs AA! and BB f be represented by the points Oco and la respectively (fig. 96). Let xx 1 be any other pair of the invo- lution. We then have, clearly from the above, xx* = a. Further, the cross ratios in question give U (Oco, 1 a) = - , B (Oco, 005') = -.. x a x' These are equal, if and only if x j = ax, or if xx f = ax*. But this implies the relation a = ax 2 , and since we have a = 0, this gives x* = 1. The only pair of the involution satisfying the conditions of the theorem is therefore the pair (7(7' = 1, a. EXERCISES 1. Show that an involution which has two harmonic conjugate pairs has double points if and only if 1 has a square root in the field. 2. Show that any involution may be represented by the equation afx a. _ .,. FIG. 90 234 ONE-DIMENSIONAL PEOJECTIVITIES [CHAP. VIII 3. The equation of Ex. 13, p. 230, is the condition that the lines joining the three pairs of points aa' ? W/, cc' on a conic are concurrent. 4. Show that if the involution x'x = a has a conjugate pair W such that the cross ratio R(0o>, W) has the value X, the number a\ has a square root in the field. 82. Parametric representation of a conic. Let a scale be established on a conic <7 2 by choosing three distinct points of the conic as the fundamental points, say, 0=0, M= oo, A= 1. Then let us establish a system of nonhomogeneous point coordinates in the plane of the conic as follows : Let - N the line OH be the x- axis, with as origin and M as <x> x (fig. 97). Let the tangents at and M to the conic meet in a point N 9 and let the tangent ON be the y-axis, with -2V" as oo y . Finally, let the point A be the point (1, 1), so that the line AN meets the #-axis in the point for which # = 1, and AM meets the ^-axis in the point for which y = 1. Now let P = X be any point on the conic. The coordinates (a?, y) of P are determined by the intersections of the lines PN and PM with the #-axis and the y-axis respectively. We have at once the relation FIG. 97 since the points 0, oo, 1, X on the conic are perspective from M with points 0, oo, 1, y on the y-axis. To determine x in terms of X, we note, first, that from the constructions given, any line through N meets the conic (if at all) in two points whose sum in the scale is 0. In par- ticular, the points 1, 1 on the conic are collinear with N and the point 1 on the #-axis, and the points X, X on the conic are collinear with N and the point x on the re-axis. Since the latter point is also on the line joining and co on the conic, the construction for multi- plication on the conic shows that any line through, the point x on 82] PAEAMETEIC BEPEESENTATION 235 the #-axis meets the conic (if at all) in two points whose product is constant, and hence equal to X 2 . The line joining the point x on the #-axis to the point 1 on the conic therefore meets the conic again in the point X 2 . But now we have 0, co, 1, X 2 on the conic perspective from the point 1 on the conic with the points 0, co, 1, $ on the #-axis. This gives the relation x = X 2 . We may now readily express these relations in homogeneous form. If the triangle OMN is taken as triangle of reference, ON being x^ = 0, 031 being os a = 0, and the point A being the point (1, 1, 1), we pass from the nonhomogeneous to the homogeneous by simply placing x = xjx v y = xjx y The points of the conic C' 2 may then le represented ly the relations (1) a^ia^ifljBssX^X,:!. This agrees with our preceding results, since the elimination of \ between these equations gives at once a: '-^053=0, which we have previously obtained as the equation of the conic. It is to be noted that the point M on the conic, which corresponds to the value X = oo, is exceptional in this equation. This exceptional character is readily removed by writing the parameter X homogene- ously XsaX^Xy Equations (1) then readily give THEOREM 25. A conic may le represented analytically ly the equa- tions x^ : X 2 : x s = \f : \\ : X 2 2 . This is called a parametric representation of a conic. EXERCISES 1. Show that the equation" of the line joining two points X 19 X 2 on the conic (1) above is x l (X x -f X 2 ) # 2 + \iX 2 x s = ; and that the equation of the tan- gent to the conic at a point \i is x l 2 A^g + X 2 # 3 = 0. Dualize. 2. Show that any collineation leaving the conic (1) invariant is of the form (Hint. Use the parametric representation of the conic and let the projectivity generated on the conic by the collineation be Xf = a\i + /JXg, X^ = y\i + 8X 2 .) CHAPTER IX GEOMETRIC CONSTRUCTIONS. INVARIANTS 83. The degree of a geometric problem. The specification of a line by two of its points may be regarded as a geometric operation* The plane dual of this operation is the specification of a point by two lines. In space we have hitherto made use of the following geometric operations : the specification of a line by two planes (this is the space dual of the first operation mentioned above) ; the specification of a plane by two intersecting lines (the space dual of the second operation above) ; the specification of a plane by three of its points or by a point and a line ; the specification of a point by three planes or by a plane and a line. These operations are known as linear operations or operations of the first degree, and the elements deter- mined by them from a set of given elements are said to be obtained by linear constrictions, or by constrictions of the first degree. The reason for this terminology is found in the corresponding analytic formulations. Indeed, it is at once clear that each of the two linear operations in a plane corresponds analytically to the solution of a pair of linear equations ; and the linear operations in space clearly correspond to the solution of systems of three equations, each of the first degree. Any problem which can be solved by a finite sequence of linear constructions is said to be a linear prollem or a problem of the first* degree. Any such problem has, if determinate, one and only one solution. In the usual representation of the ordinary real projective geometry in a plane by means of points and lines drawn, let us say, with a pencil on a sheet of paper, the linear constructions .are evidently those that can be carried out by the use of a straightedge alone. There is no familiar mechanical * An operation on one or more elements is defined as a correspondence whereby to the set of given elements corresponds an element of some sort (cf. 48). If the latter element is uniquely defined by the set of given elements (in general, the order of the given elements is an essential factor of this determination), the operation is said to be one-valued. The operation referred to in the text is then a one-valued operation defined for any two distinct points and associating with any such pair (the order of the points is in this case immaterial) a new element, viz. a line. 236 8B] DEGEEE OF A GEOMETBIC PEOBLEM 237 device for drawing lines and planes in space. But a picture (which is the section by a plane of a projection from a point) of the lines and points of intersection of linearly constructed planes may be constructed with a straight- edge (cf. the definition of a plane). As examples of linear problems we mention : (a) the determination of the point homologous with a given point in a projectivity on a line of which three pairs of homologous points are given; (6) the determination of the sixth point of a quadrangular set of which five points are given; (c) the determination of the second double point of a projectivity on a line of which one double point and two pairs of homologous points are given (this is equivalent to (&) ) ; (d) the deter- mination of the second point of intersection of a line with a conic, one point of intersection and four other points of the conic being given, etc. The analytic relations existing between geometric elements offer a convenient means of classifying geometric problems.* Confining ourselves, for the sake of brevity, to problems in a plane, a geometric problem consists in constructing certain points, lines, etc., which bear given relations to a certain set of points, lines, etc., which are sup- posed given in advance. In fact, we may suppose that the elements sought are points only ; for if a line is to be determined, it is sufficient to determine two points of this line ; or if a conic is sought, it is suffi- cient to determine five points of this conic, etc. Similar considera- tions may also be applied to the given elements of the problem, to the effect that we may assume these given elements all to be points. This merely involves replacing any given elements that are not points by certain sets of points having the property of uniquely determining these elements. Confining our discussion to problems in which this is possible, any geometric problem may be reduced to one or more problems of the following form : Given in a plane a certain finite number of points, to construct a point which shall bear to the given points certain given relations. In the analytic formulation of such a problem the given points are supposed to be determined by their coordinates (homogeneous or nonhomogeneous), referred to a certain frame of reference. The ver- tices of this frame of reference are either points contained among the given points, or some or all of them are additional points which we * The remainder of this section follows closely the discussion given in Castel- nuovo, Lezioni di geometria, Rome-Milan, Vol. I (1904), pp. 467 fl. 238 G-EOMETEIC CONSTEUCTIONS [CHAP.IX suppose added to the given points. The set of all given points then gives rise to a certain set of coordinates, which we will denote by 1, a, &, c, ,* and which are supposed known. These numbers to- gether with all numbers obtainable from them by a finite number of rational operations constitute a set of numbers, which we will call the domain, of rationality defined ly the data.^ In addition to the coordinates of the known points (which, for the sake of simplicity, we will suppose given in nonhomogeneous form), the coordinates (x, y) of the point sought must be considered. The con- ditions of the problem then lead to certain analytic relations which these coordinates x, y and a, 6, c - - must satisfy. Eliminating one of the variables, say #, we obtain two equations, /i()-o, /.fayHO, the first containing x but not y ; the second, in general, containing both x and y. The problem is thus replaced by two problems : the first depending on the solution of f (x) = to determine the abscissa of the unknown point ; the second to determine the ordinate, assum- ing the abscissa to be known. In view of this fact we may confine ourselves to the discussion of problems depending on a single equation with one unknown. Such problems may be classified according to the equation to which they give rise. A problem is said to be algebraic if the equation on which its solution depends is algebraic, i.e. if this equation can be put in the form (1) ar+a l ar~ l +ajf-*+> + a n = 0, in which the coefficients a l9 & 2 , - , a n are numbers of the domain of rationality defined by the data. Any problem which is not algebraic is said to be transcendental Algebraic problems (which alone will be considered) may in turn be classified according to the degree n of * In case homogeneous coSrdinates are used, a, &, c, . denote the mutual ratios of the coordinates of the given elements. t A moment's consideration will show that the points whose coordinates are numbers of this domain are the points obtainable from the data by linear construc- tions. Geometrically, any domain of rationality on a line may be defined as any class of points on a line which is closed under harmonic constructions ; i.e. such that if A, 2J, C are any three points of the class, the harmonic conjugate of A with respect to 3 and C is a point of the class, 83] DEGREE OF A GEOMETEIC PEOBLEM 239 the equation on which their solutions depend. We have thus problems of the first degree (already referred to), depending merely on the solution of an equation of the first degree; problems of the second degree, depending on the solution of an equation of the second degree, etc. Account must however be taken of the fact that equation (1) may be reducible within the domain K ; in other words, that the left member of this equation may be the product of two or more poly- nomials whose coefficients are numbers of K. In fact, let us suppose, for example, that this equation may be written in the form where j> 19 fa are two polynomials of the kind indicated, and of degrees n^ and n 2 respectively (n t + n z = ri). Equation (1) is then equivalent to the two equations <W*) = o> &(*) = o. Then either it happens that one of these two equations, e.g. the first, furnishes all the solutions of the given problem, in which case fa being assumed irreducible in K, the problem is not of degree n, but of degree n^ < n ; or, both equations furnish solutions of the problem, in which case fa also being assumed irreducible in K, the problem reduces to two problems, one of degree ^ and one of degree n y In speaking of a problem of the nth degree we will therefore always assume that the associated equation of degree n is irreducible in the domain of rationality defined by the data. Moreover, we have tacitly assumed throughout this discussion that equation (1) has a root ; we shall see presently that this assumption can always be satisfied by the intro- duction, if necessary, of so-called improper elements. It is important to note, however, since our Assumptions A, E, P do not in any way limit the field of numbers to which the coordinates of all elements of our space belong, and since equations of degree greater than one do not always have a root in a given field when the coefficients of the equation belong to this field, there exist spaces in which problems of degree higher than the first may have no solutions. Thus in the ordinary real projective geometry a problem of the second degree will have a (real) solution only if the quadratic equation on which it depends has a (real) root. The example of a problem of the second degree given in the next section will serve to illustrate the general discussion given above. 240 GEOMETRIC CONSTRUCTIONS [CHAP. DC 84. The intersection of a given line with a given conic. Given a conic defined, let us say, by three points A, B, and the tangents at A and B ; to find the points of intersection of a given line with this conic. Using nonhomogeneous coordinates and choosing as #-axis one of the given tangents to the conic, as y-axis the line joining the points A and J5, and as the point (1, 1) the point C, the equation of the conic may be assumed to be of the form The equation of the given line may then be assumed to be of the form The domain of rationality defined by the data is in this case K -[!,*, 2]. The elimination of y between the two equations above then leads to the equation (1) # 2 j;# 2= 0. This equation is not in general reducible in the domain K. The problem of determining the points of intersection of an arbitrary line in a plane with a given conic in this plane is then a problem of the second degree. If equation (1) has a root in the field of the geometry, it is clear that this root gives rise to a solution of the problem proposed ; if this equation has no root in the field, the problem has no solution. If, on the other hand, one point of intersection of the line with the conic is given, so that one root of equation (1), say a? ~ r, is known, the domain given by the data is and in this domain. (1) is reducible ; in fact, it is equivalent to the equation (x r + p) (a? r) = 0. The problem of finding the remaining point of intersection then depends merely on the solution of the linear equation x T + 2> = ; * There is no loss in generality in assuming this form ; for if in the choice of coordinates the equation of the given line were of the form x = c, we should merely have to choose the other tangent as se-axis to bring the problem into the form here assumed. 84,85] PKOPOSITION K 2 241 that is, the problem is of the first degree, as already noted among the examples of linear problems. It is important to note that equation (1) is the most general form of equation of the second degree. It follows that every problem of the second degree in a plane can ~be reduced to the construction of the points of intersection of an arbitrary line with a particular conic. We shall return to this later ( 86). 85. Improper elements. Proposition K 2 . We have called attention frequently to the fact that the nature of the field of points on a line is not completely determined by Assumptions A, E, P, under which we are working. We have seen in particular that this field may be finite or infinite. The example of an analytic space discussed in the Introduction shows that the theory thus far developed applies equally well whether we assume the field of points on a line to consist of all the ordinary rational numbers, or of all the ordinary real numbers, or of all the ordinary complex numbers. According to which of these cases we assume, our space may be said to be the ordinary rational space, or the ordinary real space, or the ordinary complex space. Now, in the latter we know that every number has a square root. Moreover, each of the former spaces (the rational and the real) are clearly contained in the complex space as subspaces. Suppose now that our space S is one in which not every number has a square root. In such a case it is often convenient to be able to think of our space S as forming a subspace in a more extensive space S', in which some or all of these numbers do have square roots. We have seen that the ordinary rational and ordinary real spaces are such that they may be regarded as subspaces of a more exten- sive space in the number system associated with which the square root of any number always exists. In fact, they may be regarded as subspaces of the ordinary complex space which has this property. For a general field it is easy to prove that if a v a z , -, a n are any finite set of elements of a field F, there exists a field F, containing all the elements of F, such that each of the elements a v & 2 , , a n is a square in P. This is, of course, less general than the theorem that a field P exists in which every element of F is a square, but it is sufficiently general for many geometric purposes. In the presence of Assumptions A, E, P, H it is equivalent (cf. 54) to the following statement : 242 GEOMETRIC CONSTKUCTIONS [CHAP.IX PROPOSITION K 2 . If any finite number of involutions arc given in a space S satisfying Assumptions A, E, P, there exists a space S' of which S is a subspace,* such that all the given involutions have double points in S'. A proof of this theorem will be found at the end of the chapter. The proposition is, from the analytic point of view, that the domain of rationality determined by a quadratic problem may be extended so as to include solutions of that problem. The space S' may be called an extended space. The elements of S may be called proper elements, and those of S' which are not in S may be called improper. A projec- tive transformation which changes every proper element into a proper element is likewise a proper transformation; one which transforms proper elements into improper elements, on the other hand, is called an improper transformation. Taking Proposition K 2 for the present as an assumption like A, E, P, and H , and noting that it is consistent with these other assumptions because they are all satisfied by the ordi- nary complex space, we proceed to derive some of its consequences. THEOREM 1. A proper one-dimensional projectivity without proper double elements may always be regarded in an extended space as having two improper double elements. (A, E, P, H , K 2 )f Proof. Suppose the projectivity given on a conic. If the involu- tion which belongs to this projectivity had two proper double points, they would be the intersections of the axis of the projectivity with the conic, and hence the given projectivity would have proper double points. Let S 7 be the extended space in which (K 2 ) the involution has double points. There are then two points of S' in which the axis of the projectivity meets the conic, and these are, by Theorem 20, Chap. VIII, the double points of the given projectivity. COROLLARY 1. If a line does not meet a conic in proper points, it may be regarded in an extended space as meeting it in two improper points. (A, E, P, H , K 2 ) COROLLARY 2. Every quadratic equation with proper coefficients has two roots which, if distinct, are both proper or both improper. (A, E, P, H , K 2 ) * We use the word subspa.ce to mean any space, every point of which is a point of the space of which it is a subspace. With this understanding the subspace may be identical with the space of which it is a subspace. The ordinary complex space then satisfies Proposition K 2 . t Of. Ex., p. 261. 85] PBOPOSITION K 2 243 For the double points of any projectivity satisfy an equation of the form car + (d a) x I = (Theorem 11, Cor. 4, Chap. VI), and any quadratic equation may be put into this form. THEOREM 2. Any two involutions in the same one-dimensional form have a conjugate pair in common, which may be proper or improper. (A, E, P, H , K 2 ) This follows at once from the preceding and Theorem 23, Chap. VIII. COROLLARY. In any involution there exists a conjugate pair, proper or impro%)er, which is harmonic with any given conjugate pair. (A, E, P, H , K 2 ) For the involution which has the given pair for double elements has (by the theorem) a pair, proper or improper, in common with the given involution. The latter pair satisfies the condition of the theorem (Theorem 27, Cor. 1, Chap. IV). We have seen earlier (Theorem 4, Cor., Chap. VIII) that any two involutions with double points are conjugate. Under Proposition K 2 we may remove the restriction and say that any two involutions are conjugate in an extended space dependent on the two involutions. If the involutions are on coplanar lines, we have the following : THEOREM 3. Two involutions on distinct lines in the same plane are perspective (the center of perspectivity leing proper or improper), provided the point of intersection of the lines is a double point for loth or for neither of the involutions. (A, E, P, K 3 ) Proof. If the point of intersection of the two lines be a double point of each of the involutions, let Q and E be an arbitrary pair of one involution and Q 1 and R 1 an arbitrary pair of the other involu- tion. The point of intersection of the lines QQ r and RE J is then a center of a perspectivity which transforms elements which determine the first involution into elements which determine the second. If the point is a double point of neither of the two involutions, let M be a double point of one and M r of the other (these double points are proper or else exist in an extended space S r which exists by Proposition K 2 ). Also let .Fand N* be the conjugates of in the two involutions. Then by the same argument as before, the point of intersection of the lines MM f , NN f may be taken as the center of the perspectivity. 244 GEOMETRIC CONSTRUCTIONS [CHAP, ix It was proved in 66, Chap. VII, that the equation of any point conic is of the form (1) a n x* + a,^* + a^&l + 2 a u ^ a + 2 a^x^ + 2 a 2S x 2 x s = ; but it was not shown that every equation of this form represents a conic. The line x l = contains the point (0, x v x 5 ) satisfying (1), provided the ratio x 2 : x s satisfies the quadratic equation a^p* + 2 a#B 9 x 9 + a^x* = 0. Similarly, the lines # 2 = and x z = contain points of the locus denned by (1), provided two other quadratic equations are satisfied. By Proposition K 2 there exists an extended space in which these three quadratic equations are solvable. Hence (1) is satisfied by the coordinates of at least two distinct points P, Q (proper or improper).* A linear transformation (2) evidently transforms the points satisfying (1) into points satisfying another equation of the second degree. If, then, (2) is so chosen as to transform P and Q into the points (0, 0, 1) and (0, 1, 0) respec- tively, (1) will be transformed into an equation which is satisfied by the latter pair of points, and which is therefore of the form (3) ax* + CjXfo + c z x : x s + c 8 o^ 2 = 0. If c l = 0, the points satisfying (3) lie on the two lines x l = 0, ax 1 + c 2 # 8 + c s # 2 = ; and hence (1) is satisfied by the points on the lines into which these lines are transformed by the inverse of (2). If c x 3* 0, the trans- formation ^i = < (4) a^- * Proposition K 2 has been used merely to establish the existence of points satis- fying (1). In case there are proper points satisfying (1), the whole argument can be made without K2. 85,86] PEOBLEMS OF THE SECOND DEGREE 245 transforms the points (x v x y a? 8 ) satisfying (3) into points (x[ 9 xj, x) satisfying (5) L - *-& x* + ( 6l xi + c,x>) x> = 0. But (5) is in the form which was proved in Theorem 1, Chap. VII, to be the equation of a conic. As the points which satisfy (5) are transformed by the inverse of the product of the collineations (2) and (4) into points which satisfy (1), we see that in all cases (1) repre- sents a point conic (proper or improper, degenerate or nondegenerate). This gives rise to the two following dual theorems : THEOREM 4. Every equation of the form a^x* + a^%* + a^xl + 2 a^x^ + 2 a^x z 4- 2 a 25 x 2 x B = represents a point conic (proper or improper) which may, liowever, degenerate ; and, conversely, every point conic may be represented "by an equation of this form. (A, E, P, H , K 2 ) THEOEEM 4'. Every equation of the form AX + A ^l + A **< + 2 -W 2 + 2 4As + 2 As Vs = represents a line conic (proper or improper) which, may, however, de- generate; and, conversely, every line conic may be represented "by an equation of this form. (A, E, P, H , K 2 ) 86. Problems of the second degree. "We have seen in 83 that any problem of the first degree can be solved completely by means of linear constructions ; but that a problem of degree higher than the first cannot be solved by linear constructions alone. In regard to problems of the second degree in a plane, however, it was seen in 84 that any such problem may be reduced to the problem of find- ing the points of intersection of an arbitrary line in the plane with a particular conic in the plane. This result we may state in the following form: THEOEEM 5. Any problem of the second degree in a plane may be solved by linear constructions if the intersections of every line in the plane with a single conic in this plane are assumed "known. (A, E, P, H , K 2 ) In the usual representation of the protective geometry of a real plane by means of points, lines, etc., drawn with a pencil, say, on a sheet of paper, the linear constructions, as has already been noted, are those that can be per- formed with the use of a straightedge alone. It will be shown later that any 246 GEOMETRIC CONSTEUCTIONS [CHAP.IX conic in the real geometry is equivalent protectively to a circle. The instru- ment usually employed to draw circles is the compass. It is then clear that in this representation any problem of the second degree can be solved ly means of a straightedge and compass alone. The theorem just stated, however, shows that if a single circle is drawn once for all in the plane, the straightedge alone suffices for the solution of any problem of the second degree in this plane. The discussion immediately following serves to indicate briefly how this may be accomplished. We proceed to show how this theorem may be used in the solution of problems of the second degree. Any such problem may be reduced more or less readily to the first of the following : PROBLEM 1. To find the double points of a projectivity on a line of which three pairs of homologous points are given. We may assume PA' B' C Q B A FIG. 98 that the given pairs of homologous points all consist of distinct points (otherwise the problem is linear). In accordance with Theorem 5, we suppose given a conic (in a plane with the line) and assume known the intersections of any line of the plane with this conic. Let be any point of the given conic, and with as center project the given pairs of homologous points on the conic (fig. 98). These define a projectivity on the conic. Construct the axis of this projectivity and let it meet the conic in the points P, Q. The lines OP, OQ then meet the given line in the required double points. PROBLEM 2. To find the points of intersection of a given line with a conic of which five points are given. Let A, B, C 9 D 9 H be the given points of the conic. The conic is then defined by the projectivity D(A 9 B, C)j^E(A 9 B, 0) between the pencils of lines at D and E. SEXTUPLY PEESPECTIVE TBIANGLES 247 This projectivity gives rise to a projectivity on the given line of which three pairs of homologous points are known. The double points of the latter projectivity are the points of intersection of the line with the conic. The problem is thus reduced to Problem 1. PROBLEM 3. We have seen that it is possible for two triangles in a plane to be perspective from four different centers (ef. Ex. 8, p. 105). The maximum number of ways in which it is conceivable that two triangles may be perspective is clearly equal to the number of per- mutations of three things three at a time, ie. six. The question then arises, Is it possible to construct two triangles that are perspective from six different centers? Let the two triangles be ABC and A!ffG^ and let x l = 0, x z = 0, x s = be the sides of the first opposite to A, B, C respectively. Let the sides of the second opposite to -4', B f , C' respectively be , = 0, Fas, = 0, 0. The condition for ABC = A f B r C f is that the points of intersection of A corresponding sides be collinear, ie. (i) 01-1 -F 1 -J' 1 In like manner, the condition for BCA~ A r B r C f is (2) - 1" I' -101 -V 1 From these two conditions follows -k" U -I" 1 1-10 which is the condition for CAB = A f B f C f . Hence, if two triangles are in the relations ABC = A r r C f and BCA = A'B'C J , they are also in A A the relation CAB = A r B f C f . Two triangles in this relation are said to be triply perspective (cf. Ex. 2, p. 100). The 'domain of rationality defined by the data of our problem is clearly 248 GEOMETRIC CONSTRUCTIONS [CHAP.IX Since numbers in this domain may be found which satisfy equations (1) and (2), the problem of constructing two triply perspective tri- angles is linear. The condition for ACB = A'B'C' is A (3) V-V'=Q. If relations (1), (2), and (3) are satisfied, the triangles will be per- spective from four centers. Let Jc be the common value of W and l n (3), and let I be the common value of V and W (1). Relation (2) then gives the condition k* I = 0. The relations then define two quadruply perspective triangles. The problem of constructing two such triangles is therefore still linear. If now we add the condition for CBA = A'B ! C' 9 the two triangles will, by what precedes, be perspective from six different centers. The latter condition is (4) A"J'-Z"=0. With the preceding conditions (1), (2), (3) and the notation adopted above, this leads to the condition L3 _ 78 _ I A; (/ = JL The equation IP 1 = is, however, reducible in K ; indeed, it is equivalent to -1=0, # + * + != 0. The first of these equations leads to the condition that A ( , J3 f , C r are collinear, and does not therefore give a solution of the problem. The problem of constructing two triangles that are sextuply perspective is therefore of the second degree. The equation has two roots w, w* (proper or improper and, in general,* distinct). Hence our problem has two solutions. One of these consists of the triangles 8 == 0, X L + w*% 2 + wx^= 0. * They can coincide only if the number system is such that 1 + 1 + 1 = 0; e.g. m a finite space involving the modulus 3. 86] SEXTUPLY PERSPECTIVE TRIANGLES 249 Two of the sides of the second triangle may be improper.* The points of intersection of the sides of one of these triangles with the sides of the other are the following nine points : (0, -1, 1) ( 0, w*, -w) ( 0, w, -w 2 ) (5) (-1, 0, 1) (~w\ 0, 1 ) (-to, 0, 1 ) (-1, 1, 0) ( w, -1, ) ( < -1, ) They form a configuration which contains four configurations of the kind studied in 36, Chap. IV. All triples of points in the same row or column or term of the determinant expansion of their matrix are collinear.f If one line is omitted from a finite plane (in the sense of 72, Chap. VII) having four points on each line, the remaining nine points and twelve lines are isomorphic with this configuration. EXERCISES The problems in a plane given below that are of the second degree are to be solved by linear constructions ^ with the assumption that the points of intersection of any line in the plane with a given fixed conic in the plane are known : i.e. "with a straight- edge and a given circle in tJie plane" 1. Construct the points of intersection of a given line with a conic deter- mined by (i) four points and a tangent through one of them ; (ii) three points and the tangents through two of them ; (iii) five tangents. 2. Construct the conjugate pair common to two involutions on a line. 3. Given a conic determined by five points, construct a triangle inscribed in this conic whose sides pass through three given points of the plane. * It may be noted that in the ordinary real geometry two sides of the second triangle are necessarily improper, so that in this geometry our problem has no real solution. t They all lie on any cubic curve of the form x* + xj + xj -j- 3 \xix^x 9 = for any value of X, and are, in fact, the points of inflexion of the cubic. This configura- tion forms the point of departure for a variety of investigations leading into many different branches of mathematics. 250 GEOMETRIC CONSTRUCTIONS 4. Given a triangle A^B Z C 2 inscribed in a triangle A l B 1 C l . In how many ways can a triangle A 3 B S C 3 be inscribed in A 2 B Z C 2 and circumscribed to A^C^t Show that in one case, in which one vertex of A S B^C S may be chosen arbitrarily, the problem is linear (cf. 36, Chap. IV) ; and that in another case the problem is quadratic. Show that this problem gives all COn- rt g figurations of the symbol . Give the constructions for all cases (cf. S. Ivantor, Sitzungsberichte dermathematisch-naturwissenschaftlichen Classe der Kaiserlichen Akademie der Wissenschaften zu Wien, Vol. LXXXIV (1881), p. 915). 5. If opposite vertices of a simple plane hexagon PiP 2 P s P 4 P 5 P G are on three concurrent lines, and the lines P-JP^ PJP^ Pf^ are concurrent, then the lines P 2 P 3 , PPV P Q Pi are also concurrent, and the figure thus formed is a configuration of Pappus. 6. Show how to construct a simple ?i-point inscribed in a given simple n-point and circumscribed to another given simple n-point. 7. Show how to inscribe in a given conic a simple n-point whose sides pass respectively through n given points. 8. Construct a conic through four points and tangent to a line not meeting any of the four points. 9. Construct a conic through three points and tangent to two lines not meeting any of the points. 10. Construct a conic through four given points and meeting a given line in two points harmonic with two given points on the line. 11. If A is a given point of a conic and A r , Y are two variable points of the conic such that AX, A Y always pass through a conjugate pair of a given involution on a line I, the line XY will always pass through a fixed point B. The line AB and the tangent to the conic at A pass through a conjugate pair of the given involution. 12. Given a collineation in a plane and a line which does not contain a fixed point of the collineation ; show that there is one and only one point on the line which is transformed by the collineation into another point on the line. 13. Given four skew lines, show that there are in general two lines which meet each of the given four lines ; and that if there are three such lines, there is one through every point on one of the lines. 14. Given in a plane two systems of five points A^A^A^A^A^ and BiB^B^B^ ; given also a point X in the plane, determine a point Y such that we have X (A^^A^A^A^j- F(B 1 J5 3 B 8 J5 4J B 6 ). In general, there is one and only one such point F. Under what condition is there more than one? (R. Sturm, Mathematische Annalen, Vol. I (1869), p. 533.*) * This is a special case of the so-called problem of projectivity. For references and a systematic treatment see Sturm, Die Lehre von den geometrischen Ver- wandtschaften, Vol. I, p. 348. 37] USTVABIANTS 251 87. Invariants of linear and quadratic binary forms. An expres- sion of the form a^ -h a z x 2 is called a linear "binary form in the two variables x v x 2 . The word linear refers to the degree in the variables, the word Unary to the number (two) of the variables. A convenient notation for such a form is a^ The equation defines a unique element A of a one-dimensional form in which a scale has been established, viz. the element whose homogeneous co- ordinates are (x v x 2 ] = (a v a x ). If l x = \x^ + 6 2 # 2 is another linear binary form determining the element B, say, the question arises as to the condition under which the two elements A and B coincide. This condition is at once obtained as the vanishing of the determinant A formed by the coefficients of the two forms ; i.e. the elements A and B will coincide if and only if we have A = a, a, = 0. Now suppose the two elements A and B are subjected to any pro- jective transformation II : =0. The forms a x and l x will be transformed into two forms a x , and & a ', respectively, which, when equated to 0, define the points A r 3 B r into which the points A, B are transformed by II. The coefficients of are readily calculated as the forms a x , 9 l x , in terms of those of a x) l follows: which gives Similarly, we find =a (aa l i -h Now it is clear that if the elements -4, B coincide, so also will the new elements A 1 , B 1 coincide. If we have A = 0, therefore we should also have A' = We have ,' = 0. That this is the case is readily verified. A' = 252 GEOMETEIC CONSTRUCTIONS [CHAP, ix by a well-known theorem in determinants. Tins relation may also be written a ft A' = 7 8 A. The determinant A is then a function of the coefficients of the forms a *> b*> wikh tt 16 property that, if the two forms are subjected to a lin- ear homogeneous transformation of the variables (with nonvamshing determinant), the same function of the coefficients of the new forms is equal to the function of the coefficients of the old forms multiplied by an expression which is a function of the coefficients of the trans- formation only. Such a function of the coefficients of two forms is called a (simultaneous) invariant of the forms. Suppose, now, we form the product a x - l x of the two forms a x , l a ,. If multiplied out, this product is of the form a = # 2 + 2 a Any such form is called a quadratic Unary form. Under Proposi- tion K 2 every such form may be factored into two linear factors (proper or improper), and hence any such form represents two ele- ments (proper or improper) of a one-dimensional form. These two elements will coincide, if and only if the discriminant D a = a^ a n ' a w ^ ^ e quadratic form vanishes. The condition D a =s there- fore expresses a property which is invariant under any projectivity. If, then, the form a% be subjected to a projective transformation, the discriminant D a , of the new form a r x must vanish whenever D n van- ishes. There must accordingly be a relation of the form D a , = k - D a . If &l be subjected to the transformation II given above, the coefficients new form aj are readily found to be 2 (1) < - a^aft + a u (a$ + /3y) + a By actual computation the reader may then verify the relation a /3 3 7 8 The discriminant D a of a quadratic form a 2 is therefore called an invariant of the form. 87] INVARIANTS 253 Suppose, now, we consider two binary quadratic forms Each of these (under JK 2 ) represents a pair of points (proper or im- proper). Let us seek the condition that these two pairs be harmonic. This property is invariant under protective transformations ; we may therefore expect the condition sought to be an invariant of the two forms. We know that if a v a 2 are the nonliomogeneous coordinates of the two points represented by a* = 0, we have relations = ^ss, a +a = 2 *V a n " a n with similar relations for the nonhomogeneous coordinates & 1? 6 2 of the points represented by &| = 0. The two pairs of points a v a z ; l v 6 2 will be harmonic if we have (Theorem 13, Cor. 2, Chap. VI) * = -!. L This relation may readily be changed into the following : which, on substituting from the relations just given, becomes 1 -2o 1 A.= 0. This is the condition sought. If we form the same function of the coefficients of the two forms a^ 9 V x * obtained from a*, I* by subjecting them to the transformation II, and substitute from equations (1), we obtain the relation In the three examples of invariants of binary forms thus far obtained, the function of the new coefficients was always equal to the function of the old coefficients multiplied by a power of the determinant of the transformation. This is a general theorem regard- ing invariants to which we shall refer again in 90, when a formal definition of an invariant will be given. Before closing this section, however, let us consider briefly the cross ratio B (a-fl 29 &A) f ^ e two pairs of points represented by a% = 0, J = 0. This cross ratio 254 GEOMETRIC CONSTRUCTIONS [CHAP, ix is entirely unchanged when the two forms are subjected to a pro- jective transformation. If, therefore, this cross ratio be calculated in terms of the coefficients of the two forms, the resulting function of the coefficients must be exactly equal to the same function of the coefficients of the forms a^ b f x ; the power of the determinant referred to above is in this case zero. Such an invariant is called an absolute invariant; for purposes of distinction the invariants which when transformed are multiplied by a power = of the determinant of the transformation are then called relative invariants. EXERCISES 1. Show that the cross ratio Bs (a 1 <7 2 , b^ referred to at the end of the last section is and hence show, by reference to preceding results, that it is indeed an absolute invariant. 2. Given three pairs of points defined by the three binary quadratic forms #2 Q } jj = o, c = ; show that the three will be in involution if we have H 12 22 _ Hence show that the above determinant is a simultaneous invariant of the three forms (cf. Ex. 13, p. 230). 88. Proposition K^ If we form the product of n linear binary forms a x a' x a'J a~ l \ we obtain an expression of the form An expression of this form is called a "binary homogeneous form or quant of the nth degree. If it is obtained as the product of n linear forms, it will represent a set of n points on a line (or a set of n ele- ments of some one-dimensional form). If it is of the second degree, we have, by Proposition K 2 , that there exists an extended space in which it represents a pair of points. At the end of this chapter there will be proved the following generali- zation of K 2 : 88,89] PBOPOSITION K n 255 PROPOSITION K n . If a, a l x) - are a finite number of Unary "homo- geneous forms whose coefficients are proper in a space S which satisfies Assumptions A, E, P, there exists a space S', of which S is a sub- space, in the number system of which each of these forms is a product of linear factors. As in 85, S' is called an extended space, and elements in S' but not in S are called improper elements. Proposition K n thus implies that an equation of the form a * = can always be thought of as representing n (distinct or partly coinciding) improper points in an extended space in case it does not represent any proper points. Proposition K n could be introduced as an (not independent) assump- tion in addition to A, E, P, and H . Its consistency with the other assumptions would be shown by the example of the ordinary com- plex space in which it is equivalent to the fundamental theorem of algebra, 89. Taylor's theorem. Polar forms. It is desirable at this point to borrow an important theorem from elementary algebra. DEFINITION. Given a term Ax" of any polynomial, the expression nAx?~ l is called the derivative of Ax? with respect to x t in symbols Ax* = nAx?-\ The derivative of a polynomial with respect to x i is, by definition, the sum of the derivatives of its respective terms. o This definition gives at once A == 0, if A is independent of # . u%i Applied to a term of a binary form it gives -^ With this definition it is possible to derive Taylor's theorem for the expansion of a polynomial. *We state it for a binary form as follows : Given the binary form A * For the proof of this theorem on the basis of the definition just given, cf . Fine, College Algebra, pp. 460-462. 256 GEOMETEIC CONSTRUCTIONS [CHAP.IX If herein we substitute for x v x 2 respectively the expressions x l + \y v t + *# 2 > we obtain, / n o + * Here the parentheses are differential operators. Thus where -^ means means rH etc. It is readily daf ^iL^iJ ^2^1 && 2 |_teJ proved for any term of a polynomial (and hence for the polynomial itself) that the value of such a higher derivative as d z f/dx z 8x l is independent of the order of differentiation ; i.e. that we have a 2 / = a 2 / DEFINITION. The coefficient of X in the above expansion, viz. y l df/8x 1 + 2/2#//d# 2 is called the first polar form of (y l} y z ) with respect to f (x v x z ) ; the coefficient of X 2 is called the second ; the coefficient of \ n is called the nth polar form of (y v # 2 ) ivitli respect to the form f. If any polar form be equated to 0, it represents a set of points which is called the first) second, , nth polar of the point (y 1? y 3 ) with respect to the set of points represented by f (x v # 2 ) = 0. Consider now a binary ioim.f(x v cc 2 ) = and the effect upon it of a projective transformation If we substitute these values in / (x v o? 3 ), we obtain a new form &(8l> %) A point (x v x z ) represented by/(^, o? 2 )= will be trans- formed into a point (a?/, x$ represented by the form F(x' v x)= 0. Moreover, if the point (y l9 y 3 ) be subjected to the same projectivity, it is evident from the nature of the expansion given above that the polars of (y l9 y z ) with respect to f(x v # 2 ) = are transformed into the polars of (y[, y a ') with respect to F(x[, x) = 0. 89,90] INVAKLAJSTTS 257 We may summarize the results thus obtained as follows : THEOREM 6. If a Unary form f is transformed ly a protective transformation into the form F, the set of points represented byf=Q is transformed into the set represented ly F=Q. Any polar of a point (y v y^ with respect to f = is transformed into the correspond- ing polar of the point (y[, y) with respect to J?= 0. The following is a simple illustration of a polar of a point with respect to a set of points on a line. The form x^ = represents the two points whose nonhomo- geneous coordinates are and oo respectively. The first polar of any point (y v y^) with respect to this form is clearly y^ & + y^ = 0, and represents the point ( y v y z ) ; in other words, the first polar of a point P with respect to the pair of points represented by the given form is the harmonic conjugate of this point with respect to the pair. EXERCISE Determine the geometrical construction, of the (n l)th polar of a point with respect to a set of n distinct points on a line (cf. Ex. 3, p. 51). 90. Invariants and covariants of binary forms. DEFINITION. If a binary form a = a Q x" + na l x"~~ l x z + + a> n %S be changed by the transformation , _ _ , * into a new form A = A^x[ n + A$*~' L xl H ----- h A n x n , any rational function I(a Q) a v - , a n ) of the coefficients such that we have I(A 4 V ..-, A n ) = <j>(a, A 7, 8) I(a Q> a v , a n ) is called an invariant of the form a. A function tfK^, -.,.; x lt x z ) of the coefficients and the variables such that we have C(A A v .>.,A n \ x[, xf) ^ (a, /3, 7, 8) . C(a , a v -.- 9 a n ; x v x^) is called a covariant of the form a. The same terms apply to func- tions I and C of the coefficients and variables of any finite number of binary forms with the property that the same function of the coefficients and variables of the new forms is equal to the original function multiplied by a function of a, & 7, S only ; they are then called simultaneous invariants or covariants. 258 GEOMETRIC CONSTRUCTIONS [CHAP.IX In 87 we gave several examples of invariants of binary forms, linear and quadratic. It is evident from the definition that the con- dition obtained ly equating to any invariant of a form (or of a system of forms] must determine a property of tlie set of points represented by the form (or forms) which is invariant under a pro- jective transformation. Hence the complete study of the projective geometry of a single line would involve the complete theory of invari- ants and covariants of binary forms. It is not our purpose in this book to give an account of this theory. But we will mention one theorem which we have already seen verified in special cases. Tlie functions <(#, /3, y, 8) and -^(a, /3, % 8) occurring in the definition above are always powers of tlie determinant aS /3<y of the projective transformation in question.* Before closing this section we will give a simple example of a cova- riant. Consider two binary quadratic forms a*, b* and form the new quantic c a * = K & i - <* A) x i + (<*>& - & A) ^ A + ( a A - a A) x l- By means of equations (1), 87, the reader may then verify without difficulty that the relation holds, which proves C ab to be a covariant. The two points represented by C& = are the double points (proper or improper) of the involu- tion of which the pairs determined by a* = 0, b* = are conjugate pairs. This shows why the form should be a covariant. EXERCISE Prove the statement contained in the next to the iast sentence. 91. Ternary and quaternary forms and their invariants. The remarks which have been made above regarding binary forms can evidently be generalized. A p-aryform of the nth degree is a polynomial of the nth degree homogeneous in p variables. When the number of variables is three or four, the form is called ternary or quaternary respectively. The general ternary form of the second degree when equated to zero has been shown to be the equation of a conic. In general, the set of points (proper and improper) in a plane which satisfy an equation - = * For proof, cf M for example, Grace and Young, Algebra of Invariants, pp. 21, 22. 01] INVARIANTS 259 obtained by equating to zero a ternary form of the nth degree is called an algebraic curve of the nth degree (order). Similarly, the set of points determined in space by a quaternary form of the nth degree equated to zero is called an algebraic surface of the nth degree. The definitions of invariants and covariants of jp-ary forms is pre- cisely the same as that given above for binary forms, allowance being made for the change in the number of variables. Just as in the binary case, if an invariant of a ternary or quaternary form vanishes, the corresponding function of the coefficients of any protectively equivalent form also vanishes, and consequently it represents a prop- erty of the corresponding algebraic curve or surface which is not changed when the curve or surface undergoes a projective transforma- tion. Similar remarks apply to covariants of systems of ternary and quaternary forms. Invariants and covariants as defined above are with respect to the group of all projective collineations. The geometric properties which they represent are properties unaltered by any projective collineatiou. Like definitions can of course be made of invariants with respect to any subgroup of the total group. Evidently any function of the coefficients of a form which is invariant under the group of all col- lineations will also be an invariant under any subgroup. But there will in general be functions which remain invariant under a subgroup but which are not invariant under the total group. These correspond to properties of figures which are invariant under the subgroup with- out being invariant under the total group. We thus arrive at the fundamental notion of a geometry as associated with a given group, a subject to which we shall return in detail in a later chapter. EXERCISES 1. Define by analogy with the developments of 80, the n - 1 polars of a ternary or quaternary form of the nth degree. 2. Regarding a triangle as a curve of the third degree, show that the second polar of a point with regard to a triangle is the polar line defined on page 46. 3. Generalize Ex. 2 in the plane and in space, and dualize. ^11 ^12 ^13 4. Prove that the discriminant a la a 22 a 23 of the ternary quadratic form is an invariant. What is its geometrical interpretation? Cf. Ex., p. 187. 260 GEOMETEIC CONSTRUCTIONS [CHAP. IX 92. Proof of Proposition K n . Given a rational integral function <f> (x) = a tf + a^"- 1 -{ ----- h a n , a^ 0, whose coefficients belong to a given field F, and which is irreducible in F, there exists a field F', containing F, in which the equation <f> (x) = has a root. Let f(x) be any rational integral function of x with coefficients in F, and let / be an arbitrary symbol not an element of F. Consider the class F^ = [/(/)] of all symbols f(j)> where [/(&)] is the class of all rational integral functions with coefficients in F. We proceed to define laws of combination for the elements of Fj which render the latter a field. The process depends on the theorem * that any poly- nomial f(x) can be represented uniquely in the form where q(x) and r(x) are polynomials belonging to F, i.e. with coefficients in F, and where r(x) is of degree lower than the degree n of <j> (x). If two polynomials f l9 / 2 belonging to F are such that their difference is exactly divisible by <f>(x), then they are said to be congruent modulo <f>(x), in symbols / t = / 2 , mod. <p(x). 1. Two elements /^j), / 2 (/) of F, are said to be equal, if and only if/!^) andj^as) are congruent mod. <p(x). By virtue of the theorem referred to above, every element /(/) of F y is equal to one and only one element f f (j) of degree less than n. We need hence consider only those elements f(j) of degree less than n. Further, it follows from this definition that <f> (/) = 0. 2. If /!(*)+/, (a) s/ t (a), mod * (as), then/ 1 (y)+/ 2 (y)-/ 8 (/). 3. If /, (x) /, (x) s/ 8 (*), mod. * (x), then / t (/) ./, (/) =/ 3 (/). Addition and multiplication of the elements of F y having tlius been defined, the associative and distributive laws follow as immedi- ate consequences of the corresponding laws for the polynomials /(#). It remains merely to show that the inverse operations exist and are unique. That addition has a unique inverse is obvious. To prove that the same holds for multiplication (with the exception of 0) we need only recall f that, since <(#) and any polynomial f(x) have no common factors, there exist two polynomials h(x) and Jc(x) with coefficients in F such that A (a?) -/() + ft (a?). ^ () = !. * Fine, College Algebra, p. 156. f Fine, loc. cit., p. 208. 92] PEOOF OF K n 261 This gives at once h (/) /(/) = 1, so that every element /(/) distinct from has a reciprocal. The class F y is therefore a field with respect to the operations of addition and mutiplication defined above (cf. 52), such that <(/)= 0. It follows at once* that xj is a factor of $(x) in the field F, which is there- fore the required field F'. The quotient <f>(x)/(xj) is either irre- ducible in Fj, or, if reducible, has certain irreducible factors. If the degree of one of the latter is greater than unity, the above process may be repeated leading to a field F^,,/' being a zero of the factor in question. Continuing in this way, it is possible to construct a field ^7,y,...,/ m) > w ^ ere fl&^w 1, in which <f>(x) is completely reducible, i.e. in which <(#) may be decomposed into n linear factors. This gives the following corollary : Given a polynomial <f> (x) "belonging to a given field F, there exists a field F 7 containing F in which <j> (x) is completely reducible. Finally, an obvious extension of this argument gives the corollary : Given a finite number of polynomials each of which belongs to a given field F, there exists a field F', containing F, in which each of the given polynomials is completely reducible. This corollary is equivalent to Proposition K n . For if S be any space, let F be the number system on one of its lines. Then, as in the Introduction (p. 11), F' determines an analytic space which is the required space S' of Proposition K n . The more general question at once presents itself: Given a field F, does there exist a field F, containing F, in which every polynomial belonging to F is completely reducible ? The argument used above does not appear to offer a direct answer to this question. The ques- tion has, however, recently been answered in the affirmative by an extension of the above argument which assumes the possibility of " well ordering " any class, f EXERCISE Many theorems of this and other chapters are given as dependent on A, E, P, H , whereas they are provable without the use of H . Determine which theorems are true in those spaces for which H is false. * Fine, College Algebra, p. 1G9. t Cf. E. Steinitz, Algebraische Theorie der Kbrper, Journal fur reine u. ange- wandte Mathematik, Yol. CXXXVII (1909), p. 167 ; especially pp. 271-286. CHAPTER PROJECTIVE TRANSFORMATIONS OF TWO-DIMENSIONAL FORMS 93. Correlations between two-dimensional forms. DEFINITION. A projective correspondence between the elements of a plane of points and the elements of a plane of lines (whether they be on the same or on different bases) is called a correlation. Likewise, a projective correspondence between the elements of a bundle of planes and the elements of a bundle of lines is called a correlation.^ Under the principle of duality we may confine ourselves to a con- sideration of correlations between planes. In such a correlation, then, to every point of the plane of points corresponds a unique line of the plane of lines ; and to every pencil of points in the plane of points corresponds a unique projective pencil of lines in the plane of lines. In particular, if the plane of points and the plane of lines are on the same base, we have a correlation in a planar field, whereby to every point P of the plane corresponds a unique line p of the same plane, and in which, if Jf, P& P& P are collinear points, the corresponding lines p v p 2 , p z) p^ are concurrent and such that That a correlation T transforms the points [P] of a plane into the lines [p] of the plane, we indicate as usual by the functional notation The points on a line I are transformed by F into the lines on a point L. This determines a transformation of the lines [I] into the points [], which we may denote by T', thus: T'(l) = L. That F ; is also a correlation is evident (the formal proof may be supplied by the reader). The transformation T r is called the correla- tion induced by T. If a correlation T transforms the lines [I] of a * All developments of this chapter are on the basis of Assumptions A, E, P, and H . Cf . the exercise at the end of the last chapter. t The terms reciprocity and dudlity are sometimes used in place of correlation. 262 93] CORRELATIONS 263 plane into the points [L] of the plane, the correlation which trans- forms the points [ll r ] into the lines [LL f ] is the correlation induced by T. If F is induced by F, it is clear that T is induced by P. For if we have we have also and hence the induced correlation of F transforms P 2 into p^ etc. That correlations in a plane exist follows from the existence of the polar system of a conic. The latter is in fact a projective transforma- tion in which to every point in the plane of the conic corresponds a unique line of the plane, to every line corresponds a unique point, and to every pencil of points (lines) corresponds a projective pencil of lines (points) (Theorem 18, Cor., Chap. V). This example is, how- ever, of a special type having the peculiarity that, if a point P corre- sponds to a line p, then in the induced correlation the line p will correspond to the point P ; i.e. in a polar system the points and lines correspond doubly. This is by no means the case in every correlation. DEFINITION. A correlation in a plane in which the points and lines correspond doubly is called a polarity. It has been found convenient in the case of a polarity defined by a conic to study a transformation of points into lines and the induced transformation of lines into points simultaneously. Analogously, in studying collineations we have regarded a transformation T of points P v P^ P B , P into points _Zf, P^ PJ 9 P^ and the transformation T' of the lines P& P 2 P B , P B P^ 1J1> into the lines IJ'JJ', PjPj, #2J', #?' as the same collineation. In like manner, when considering a trans- formation of the points and lines of a plane into its lines and points respectively, a correlation F operating on the points and its induced correlation T 1 operating on the lines constitute one transformation of the points and lines of the plane. For this sort of transformation we shall also use the term correlation. In the first instance a correlation in a plane is a correspondence between a plane of points (lines) and a plane of lines (points). In the extended sense it is a transformation of a planar field either into itself or into another planar field, in which an element of one kind (point or line) corresponds to an ele- ment of the other kind. 264 TWO-DIMENSIONAL PBOJECTIVITIES [CHAP.X The following theorem is an immediate consequence of the defini- tion and the fact that the resultant of any two protective correspond- ences is a projective correspondence. THEOREM 1. The resultant of two correlations is a projective col- lineation, and the resultant of a correlation and a projective collinea- tion is a correlation. We now proceed to derive tins fundamental theorem for correlations between two-dimensional forms. THEOREM 2. A correlation letween two two-dimensional primitive forms is uniquely defined when four pairs of homologous elements are given, provided that no three elements of either form are on the same one-dimensional primitive form. Proof. Let the two forms be a plane of points a and a plane of lines #'. Let C* be any conic in a r , and let the four pairs of homol- ogous elements be A 9 3, C, D in a and a', V, c', dl in a!. Let A!, B f , C\ D f be the poles of a f , V, c f , d r respectively with respect to C 2 . If the four points A, B, C, D are the vertices of a quadrangle and the four points A', B r , C r , D* are likewise the vertices of a quadrangle (and this implies that no three of the lines a r , V, c r , d f are concurrent), there exists one and only one collineation transforming A into A r , B into B f , C into C 1 , and D into D f (Theorem 18, Chap. IV). Let tins collineation be denoted by T, and let the polarity defined by the conic C 2 be denoted by P. Then the projective transformation F which is the resultant of these two transforms A into a r , B into V, etc. More- over, there cannot be more than one correspondence effecting this transformation. For, suppose there were two, F and F r Then the projective correspondence I^" 1 F would leave each of the four points A, B, C, D fixed; i.e. would be the identity (Theorem 18, Chap. IY). But this would imply F x = F. THEOREM 3. A correlation which interchanges the vertices of a triangle with the opposite sides is a polarity. Proof. Let the vertices of the given triangle be A, B, C, and let the opposite sides be respectively a, I, c. Let P be any point of the plane ABC which is not on a side of the triangle. The line p into which P is transformed by the given correlation F does not, then, pass through a vertex of the triangle ABC. The correlation F is deter- mined by the equation F (ABCP) = alcp, and, by hypothesis, is such 93] COEKELATIONS 265 that F (ale) ABC. The points [Q] of c are transformed into the lines [g] on C, and these meet c in a pencil [Q r ] projective with [Q] (fig. 99). Since A corresponds to B and to A in the projectivity [Q] ~^[Q f ], this projectivity is an involution L The point Q in which CP meets c is transformed by F into a line on the point cp\ and since Q and cp are paired in I, it follows that cp is transformed into the line CQ Q CP. In like manner, Ip is transformed into BP. Hence p = (cp, lp] is transformed into P = (CP, BP). THEOBEM 4. Atiy projective collineatwn, II, m a plane, <x, is the product of two polarities. Proof, Let Aa be a lineal element o! a, and let H (4a) =* jl V, n (4W) A" a". Unless II is perspective, Aa may be so chosen that A 9 A f , A n are not collinear, aa r a fr are not concurrent, and no line of one of the three lineal elements passes through the point of another. In this case there exists a polarity P such that J?(AA'A rr ) = a"a'a, namely the polarity- defined by the conic with regard to which AA"(aa ff ) is a self-polar tri- angle and to which o! is tangent at A f . If II is perspective,,the existence of P follows directly on choosing Aa, so that neither A nor a is fixed. We then have and hence the triangle AA[(aaf) is self-reciprocal. Hence (Theorem 3) PIT ss Pj is a polarity, and therefore II = PP r 266 TWO-DIMENSIONAL PBOJECTIVITIES [CHAI>. x 94. Analytic representation of a correlation between two planes. Bilinear forms. Let a system of simultaneous point-and-line coordi- nates be established in a planar field. We then have THEOREM 5. Any correlation in a plane is given as a transforma- tion of points into lines by equations of the form (1) pul = a^ + where the determinant A of the coefficients a l} is different from zero. Conversely, every transformation of this form in which the determinant A is different from zero represents a correlation. The proof of this theorem is completely analogous to the proof of Theorem 8, Chapter VII, and need not be repeated here. As a corollary we have COROLLARY 1. The transformation pu[~x v pu^x^ pu! } =x s in a plane represents a polarity in which to every side of the triangle of reference corresponds the opposite vertex. Also, if (u[, ii' 2) u^) be interpreted as line coordinates in a plane different from that containing the points (x lf # 2 , x s ) (and if the num- ber systems are so related that the correspondence X 1 = X between the two planes is projective), we have at once COROLLARY 2. The equations of Theorem 5 also represent a correla- tion between the plane of (x v w z > #3) and the plane of (%', ul, v. 8 '). Returning now to the consideration of a correlation in a plane (planar field), we have seen that the equations (1) give the coordi- nates (u[, tf a ', ui) of the line u'= T (X), which corresponds to the point -3T= (x v % z , a? 8 ). By solving these equations for x i9 we obtain the coordinates of Z= T" 1 (vf) in terms of the coordinates u[ of the line to which Xis homologous in the inverse correlation T"\ If, however, we seek the coordinates of the point -5T' = T (u] which corresponds to any line u in the correlation T, we may proceed as follows : COKBELATI02TS 267 Let the equation of the point JT' = (&[, x! 2 , x' A ) in line coordinates be = 0. Substituting in this equation from (1) and arranging the terms as a linear expression in x v x 2) # 3 , = 0, we readily find (3) The coordinates of X ! in terms of the coordinates of u are then given by (4) This is the analytic expression of the correlation as a transformation of lines into points ; i.e. of the induced correlation of F. These equa- tions clearly apply also in the case of a correlation between two different planes. It is perhaps well to emphasize the fact that Equations (1) express T as a transformation of points into lines, while Equations (4) represent the induced correlation of lines into points. Since we consider a correlation as a trans- formation of points into lines and lines into points, T is completely represented by (1) and (4) taken together. Equations (2) and (3) taken together repre- sent the inverse of T. Another way of representing F analytically is obtained by observ- ing that the point (x v x 2 , # 8 ) is transformed by F into the line whose equation in current coordinates (x[, x, x) is or, (5) (a^ + a 12 2 = 0. The left-hand member of (5) is a general ternary bilinear form. We have then COROLLARY 3. Any ternary bilinear form in which the determinant A is different from zero represents a correlation in a plane. 268 TWO-DIMENSIONAL PEOJECTIVITIES [CHAP.X 95. General projective group. Representation by matrices. The general protective group of transformations in a plane (which., under duality, we take as representative of the two-dimensional primitive forms) consists of all projective collineations (including the identity) and all correlations in the plane. Since the product of two collinea- tions is a collineation, the set of all projective collineations .forms a subgroup of the general group. Since, however, the product of two correlations is a collineation, there exists no subgroup consisting entirely of correlations.* According to the point of view developed in the last chapter, the projective geometry of a plane is concerned with theorems which state properties invariant under the general projective group in the plane. In particular, the principle of duality may be regarded as a consequence of the presence of correlations in this group. Analytically, collineations and correlations may be regarded as aspects of the theory of matrices. The collineation may be conveniently represented by the matrix A of the coefficients a t/ : The product of two collineations A = (a v ) and B = (&J is then given by the product of their matrices : the element of the ith row and the yth column of the matrix BA being obtained by multiplying each element of the ith row of B by the corresponding element of the^th column of A and adding the products thus obtained. It is clear that two collineations are not in general commutative. * A polarity and the identity fcym a group ; but this forms no exception to the statement just made, since the identity must be regarded as a collineation. MATEICES 269 Of the two matrices / a il <*12 13\ /*n u ai\ ^31 ^32 <W \^13 ^23 ^33/ either of which is obtained from the other by interchanging rows and columns, one is called the conjugate or transposed matrix of the other. The matrix / A A 4 \ I -fX-. i L n i uTjL - I is called the adjoint matrix of the matrix A. The adjoint matrix is clearly obtained by replacing each element of the transposed matrix by its cof actor. Equations (2) of 67 show that the adjoint of a given matrix represents the inverse of the collineation represented "by the given matrix. Indeed, by direct multiplication, i\ Kl ^22 ^23 MlS 4*2 As H and the matrix just obtained clearly represents the identical col- lineation. Since, when a matrix is thought of as representing a collineation, we may evidently remove any common factor from all the elements of the matrix, the latter matrix is equivalent to the so-called identical matrix,* II 0\ 1 . \0 I/ Furthermore, Equations (3), 67, show that if a given matrix represents a collineation in point coordinates, the conjugate of the adjoint matrix represents the same collineation in line coordinates. Also from the representation of the product of two matrices just derived, ' follows the important result: The determinant of the product of two matrices (collineations) is equal to the product of the determinants of the two matrices (col- lineations). * In the general theory of matrices these two matrices are not, however, re- garded as the same. It is only the interpretation of them as collineations which renders them equivalent. 270 TWO-DIMENSIONAL PEOJECTIVITIES [CHAP.X From what has just been said it is clear that a matrix does not completely define a collineatiou, unless the nature of the coordinates is specified. If it is desired to exhibit the coordinates in the nota- tion, we may write the collineation ?/ = 20 v a ; in the symbolic form The matrix (a v ) may then be regarded as an operator transforming the coordinates x=s(x l9 # a , # 8 ) into the coordinates #' = (#/, h # 8 ). If we place a v = a jl9 the matrix conjugate to (aj is (a,,). Also by plac- ing JT = -4,,, the adjoint matrix of (c^) is ( Jj. The inverse of the above collineation is then written Furthermore, the collineation #'= (a v )# is represented in line coordi- nates by the equation This more complete notation will not be found necessary in gen- eral in the analytic treatment of collineations, when no correlations are present, but it is essential in the representation of correlations by means of matrices. The correlation (1) of 94 may clearly be represented symbolically by the equation '(0 t ,)0, where the matrix (a^) is to be regarded as an operator transforming the point & into the line vf. This correlation is then expressed as a transformation of lines into points by The product of two correlations u f ~ (a iy ) x and %'= (B v ) x is there- fore represented by d**(Bv)(av)x (d. Equations (4), 94), or by Also, the inverse of the correlation u 1 = (a ) x is given by ^ = (l v )< or by TYPES OF COLLISrEATIOlSrS 271 EXERCISE Show that if [II] is the set of all collineations in a plane and I\ is any correlation, the set of all correlations in the plane is [Orj, so that the two sets of transformations [II] and [Iirj comprise the general projective group in the plane. By virtue of this fact the subgroup of all projective collineations is said to be of index in the general projective group.* 96. Double points and double lines of a collineation in a plane. Referring to Equations (1) of 67 we see that a point (x v x z , x s ) which is transformed into'itself by the collineation. (1) must satisfy the equations which, by a simple rearrangement, may be written (1) If a point (x v C 2 , # 3 ) is to satisfy these three equations, the deter- minant of this system of equations must vanish ; i.e. p must satisfy the equation (2) = 0. Tliis is an equation of the third degree in p, which cannot have more than three roots in the number system of our geometry. Suppose that /> t is a root of this equation. The system of equa- tions (1) is then consistent (which means geometrically that the three lines represented by them pass through the same point), and the point determined by any two of them (if they are independent, i.e. if they do not represent the same line) is a double point Solving the first two of these equations, for example, we find as the coordi- nates (% v x z> o? 3 ) of a double point (3) iv-v^s 8 * 2 Pi *u-/>i & 21 *13 * A subgroup [II] of a group is said to he of index n, if there exist n 1 trans- formations T l (i ss 1, 2, . . . n - l),-such that the n - 1 sets [nTf] of transformations together with the set [II] contain all the transformations of the group, while no two transformations within the same set or from any two sets are identical. 272 TWO-DIMENSIONAL PEOJECTIVITIES [CHAP, x which represent a unique point, unless it should happen that all the determinants on the right of this equation vanish. Leaving aside this possibility for the moment, we see that every root of Equation (2), which is called the characteristic equation of the collineatioii (or of the representative matrix), gives rise to a unique double point. Moreover, every double point is obtainable in this way. This is the analytic form of the fact already noted, that a collineation which is not a Jiomology or an elation cannot have more than three do Me points, unless it is the identical collineation. If, however, all the determinants on the right in Equations (3) vanish, it follows readily that the first two of Equations (1) represent the same line. If the determinants formed analogously from the last two equations do not all vanish, we again get a unique double point ; but if the latter also vanish, then all three of the equations above represent the same line. Every point of this line is then a double point, and the collineation must be a homology or an elation. Clearly this can happen only if p l is at least a double root of Equation (2) ; for we know that a perspective collineation cannot have more than one double point which is not on the axis of the collineation. A complete enumeration of the possible configurations of double points and lines of a collineation can be made by means of a study of the characteristic equation, making use of the theory of elementary divisors.* It seems more natural in the present connection to start with the existence of one fixed point (Proposition K 8 ) and discuss geometrically the cases that can arise. By Theorem 4 a collineation is the product of two polarities. Hence, any double point has the same polar line in both polarities, and that polar line is a double line. Hence the invariant figitre of douUc points and lines is self-dual. Four points of the plane, no three of which are collineor, cannot be invariant unless the collineation reduces to the identity. If three noncollinear points are invariant, two cases present themselves. If the collineation reduces to the identity on no side of the invariant triangle, the collineation is of Type I (cf. 40, Chap. IV). If the collineatioia is the identity on one and only one side of the invariant triangle, the collineation is of Type JF.f If two distinct points are * Cf. Baeher, Introduction to Higher Algebra, Chaps. XX and XXI. t If it is the identity on more than one side, it is the identical collineation. no] TYPES OF COLLDTEATIONS 273 invariant, but no point not on the line I joining these two is invariant, two possibilities again arise. If the collineation does not leave every point of this line invariant, there is a unique other line through one of these points that is invariant, since the invariant figure is self-dual. The collineation is then of Type II. If every point of the line is invariant, on the other hand, all the lines through a point of the line I must be invariant, since the figure of invariant elements is self-dual. The collineation is then of Type V. If only one point is fixed, only one line can be fixed. The collinea- tion is then parabolic both on the line and on the point, and the collineation is of Type III. We have thus proved that every collineation different from the identity is of one of the five types previously enumerated. Type I may be represented by the symbol [1, 1, 1], the three 1's denoting three distinct double points. In Type IV there are also three distinct double points, but all points on the line joining two of them are fixed and Equation (1) has one double root. Type 7Fis denoted by [(1, 1), 1]. In Type //, as there are only two distinct double points, Equation (1) must have a double root and one simple root. This type is ac- cordingly denoted by the symbol [2, 1], the 2 indicating the double point corresponding to the double root. Type Fis then naturally repre- sented by [(2, 1)], the parentheses again indicating that every point of the line joining the two points is fixed. Type III corresponds to a triple root of (1), and may therefore be denoted by [3]. We have then the following : THEOREM 6. Every protective collineation in a plane is of one of the following five types : [1,1,1] [(1.1). 1] [2,1] [(2, 1)] [3] In this table the first column corresponds to three distinct roots of the characteristic equation, the second column to a double root, the third column to a triple root. The first row corresponds to the cases in which there exist at least three double points which are 274 TWO-DIMENSIONAL PBOJEOTIVITIES [CHAP, x not collinear ; tlie second row to the case where there exist at least two distinct double points and all such points are on the same line ; the third row to the case in which there exists only a single -double point. With every collineation in a plane are associated certain projec- tivities on the invariant lines and in the pencils on the invariant points. In case the collineation is of Type /, it is completely deter- mined if the projectivities on two sides of the invariant triangle are given. There must therefore be a relation between the projectivities on the three sides of the invariant triangle (cf. Ex. 5, p. 276). In a collineation of Type II the projectivity is parabolic on one of the invariant lines but not on the other. The point in which the two invariant lines meet may therefore be called singly parabolic. The collineation is completely determined if the projectivities on the two invariant lines are given. In a collineation of Type /// the pro- jectivity on the invariant line is parabolic, as likewise the projectivity on the invariant point. The fixed point may then be called doubly parabolic. The projectivities on the invariant lines of a collineation of Type V are parabolic except the one on the axis which is the identity. The center is thus a singly parabolic point. In the table of Theorem 6 the symbols 3, 2, and 1 may be taken to indicate doubly and singly and nonparabolic points respectively.* We give below certain simple, so-called canonical forms of the equations defining collineations of these five types. Type L Let the invariant triangle be the triangle of reference. The collineation is then given by equations of the form in which a lv # 22 , & 33 are the roots of the characteristic equation and must therefore be all distinct. Type IV, Homology. If the vertices of the triangle of reference are taken as invariant points, the equations reduce to the form written above; but since one of the lines 3^= 0, # 3 = 0, # 3 = is pointwise * For a more detailed discussion of collineations, reference may "be made to Newson, A New Theory of Collineations, etc., American Journal of Mathematics, Vol. XXIV, p. 109. J] TYPES OF COLLIKEATIONS 275 invariant, we must have either a 22 = a ss or a ss = a n or & n = a Z2 . Thus the homology, may be written A harmonic homology or reflection is obtained by setting & 83 = 1. Type IL The characteristic equation has one double root, p^ = p , say, and a simple root /3 3 . Let the double point corresponding to Pi Pz ^ e ^i (0> 0, 1), let the double point corresponding to p z be ?7g = (1, 0, 0), and let the third vertex of the triangle of reference be any point on the double line u z corresponding to /> 3 , which line will pass through the point U r The collineation is then of the form since the lines ^=0 and a? 2 = are double lines and (1, 0, 0) is a double point. The characteristic equation of the collineation is clearly and since this must have a double root, it follows that two of the numbers a n> 22 , a 33 must be equal. To determine which, place p = a 00 ; using the minors of the second row, we find, as coordinates of the corresponding double point, (0, (a n - a 23 ) (a 22 - a S3 ), a 32 (a n - a M ) ), which is U v and hence we have #. 22 =ft 83 . The collineation then is of Type II, if a xl ^ a 2Z . Its equations are therefore where a 32 3=- and a n = a 22 . jf^e /JJ. The characteristic equation has a triple root, p 1 = p 2 = P z say. Let U : = (0, 0, 1) be the single double point, and the line x l = be the single double line. With this choice of coordinates the collineation has the form , P x i a 'n^i> 276 TWO-DIMENSIONAL PROJECTIVITIES [OHAP.X By writing the characteristic equation we find, in view of the fact that the equation has a triple root, that a n =^ 25J =^ 33 . The form of the collineation is therefore where the numbers a 21 , a sz must be different from 0. * Type F, Elation. Choosing (0, 0, 1) as center and ^=0 as axis, the equations of the collineation reduce to the form given for Type ///, where, however, a 33 must be zero in order that the line 0^= be pointwise invariant. The equations for Type II also yield an elation in case & n = a 22 . Thus an elation may be written EXERCISES 1. Determine the collineation which transforms the points A = (0, 0, 1), B = (0, 1, 0), C= (1, 0, 0), D = (1, 1, 1) into the points E, C, D, A respec- tively. Show that the characteristic equation of this collineation is (p I) (p 2 + 1) = 0, which in any field has one root. Determine the double point and double line corresponding to this root. Assuming the field of numbers to be the ordinary complex field, determine the coordinates of the remaining two double points and double lines. Verify, by actually multiplying the matrices, that this collineation is of period 4 (a fact which is evident from the defini- tion of the collineation). 2. With the same coordinates for A, JS, C, D determine the collincation which transforms these points respectively into the points Z>, A , 7>, C. The resulting collineation must, from this definition, be a homology. Why? De- termine its center and its axis. By actual multiplication of the matrices verify that its square is the identical collineation. 3. Express each of the collineations in Exs. 1 and 2 in terms of line coordinates. 4. Show that the characteristic cross ratios of the one-dimensional projec- tivities on the sides of the invariant triangle of the collineation x'z=ax v x^ = bx z , z = c# 3 are the ratios of the numbers , /;, c. Hence show that tho product of these cross ratios is equal to unity, the double points being taken around the triangle in a given order. 5. Prove the latter part of Ex. 4 for the cross ratios of tho projectivities on the sides of the invariant triangle of any collineation of Type /. TYPES OF COLLINEATIOjtfS 277 6. Write the equations of a collineation of period 3;4;5;-..;n;-... 7. By properly choosing the system of nonhomogeneous coordinates any collineation of Type / may be represented by equations x' = ax, y' = ly. The set of all collineations obtained by giving the parameters a, b all possible values forms a group. Show that the collineations #' = axj ?/' = a r y, where r is constant for all collineations of the set, form a subgroup. Show that every collineation of this subgroup leaves invariant every curve whose equation is y = cx r 9 where c is any constant. Such curves are called path curves of the collineations. 8. If P is any point of a given path curve, p the tangent at P, and A y B, C the vertices of the invariant triangle, then & (p, PA, PJ3, PC) is a constant. 9. For the values r = 1, 2, -| the path curves of the collineations of the subgroup described in Ex. 7 are conies tangent to two sides of the invariant triangle at two vertices. 10. If r = 0, the subgroup of Ex. 7 consists entirely of homologies. 11. Prove that any collineation of Type / may be expressed in the form x' = k(ax + by), / = *(& -ay), with the restriction a 2 + b 2 = 1. 12. Prove that any collineation can be expressed as a product of collinea- tions of Type /. 13. Let the invariant figure of a collineation of Type // be A, JB, I, m, where I = AB, B = Im. The product of such a collineation by another of Type // with invariant figure A', B, Z, m' is in general of Type //, but may be of Types /// or 1 V. Under what conditions do the latter cases arise ? 14. Using the notation of Ex. 13, the product of a collineation of Type II with invariant figure A, J5, Z, m by one with invariant figure -4,,J5', Z, m' is in general of Type II, but may be of Types /// or IV. Under what conditions do the latter cases arise ? 15. Prove that any collineation can be expressed as a product of collinea- tions of Type II. 16. Two collineations of Type III with the same invariant figure are not in general commutative. 17. Any protective collineation can be expressed as a product of collinea- tions of Type ///. 18. If II is an elation whose center is C, and P any point not on the axis, then P and C are harmonically conjugate with respect to n~ 1 ( J P) andH(P). 19. If two coplanar conies are projective, the correspondence between the points of one and the tangents at homologous points of the other determines a correlation* 20. If in a collineation between two distinct planes every point of the line of intersection of the planes is self-corresponding, the planes are per- spective. 278 TWO-DIMENSIONAL PROJEOTIV1TIES [CHAP, x 21. In nonhomogeneous coordinates a collineation of Type / with fixed points (o lf a a ), (b l9 6 a ) (c v c 2 ) may be written a; y 1 x y 1 ! #2 1 t ! a 2 1 cr 2 (? 1 c . 2 ^ fc'c l c ~ I /j'^ Type II may be w x y 1 ' ^ a? y 1 a x ij 1 1 ! n a 1 1 b 1% 1 /J />i & 2 1 ^* Cj c 2 1 k' ci c z 1 /t' ritten ar y 1 2: y 1 i /i ft ~\ ft /^ /; 3 1 ^j />! ^ 2 1 AV> a and Type III may f __ a? y 1 ?/ a,- y 1 i i 11 #! 2 1 1 rtj_ a 2 J. J. Z^ J 2 1 A' bi h n 1 X: ! 5 2 f *! 4 t be written 2- y 1 w i 20 s (a* 2 + 2j8J)a 1 + 2as 1 ? + W' 1 a; y 1 ^ x w a 0^ + 2^ a; y 1 ?^ w a (a 2 + 2^) 3 + 2a.V + ?a> y x y 1 &* 1 3 o u\ w 9 ai 2 +2/fr 97. Double pairs of a correlation. We inquire now regarding the existence of double pairs of a correlation in a plane. By a double pair is meant a point X and a line u such that the correlation transforms X into u and also transforms u into X; in symbols, if F is the cor- relation, such that r(Z)=w and F(w)=:X. We have already seen (Theorem 3) that if the vertices and opposite sides of a triangle are double pairs of a correlation, the correlation is a polarity. We may note first that the problem of finding the double pairs of a correlation is in one form equivalent to finding the double elements 07] DOUBLE PAIRS OF A CORRELATION 279 of a certain eollineation. In fact, a double pair X, u is such that F(JL")=^ and r 2 (J5T) = r(w)=JT, so that the point of a double pair of a correlation F is a double point of the coUineation F 2 Similarly, it may be seen that the lines of the double pairs are the double lines of the eollineation F 2 . It follows also from these considerations that F is a polarity, if F 2 is the identical eollineation. Analytically, the problem of determining the double pairs of a correlation leads to the question : For what values of (x v # 2 , # 8 ) are the coordinates of the line to which it corresponds proportional to the coordinates a x + of the line which corresponds to it in the given correlation ? If p is the unknown factor of proportionality, this condition is expressed by the equations K - P a u) x i + (* - P a n) + ( a is - P a ^ x* = 3 pa S2 ) ^ 8 = 0, which must be satisfied by the coordinates (x lf # 2 , a? 8 ) of any point of a double pair. The remainder of the treatment of this problem is* similar to the corresponding part of the problem of determining the double elements of a eollineation ( 96). The factor of proportionality p is determined by the equation *p a *l (2) which is of the third degree and has (under Proposition K 2 ) three roots, of which one is 1, and of which the other two may be proper or improper. Every root of this equation when substituted for p in (1) renders these equations consistent. The coordinates (x v x v # 8 ) are then determined by solving two of these. If the reciprocity in question is a polarity, Equations (1) must be satisfied identically, i.e. for every set of values (x v # 2 , x 3 ). This would imply that all the relations S-/>^-0 (*,/=!, 2, 3) are satisfied. 280 TWO-DIMENSIONAL PEOJECTIVITIES [CHAP. X Let us suppose first that at least one of the diagonal elements of the matrix of the coefficients (a i} ) be different from 0. If this be a lv the relation & u -/^n O gives at once /> = !; and this value leads at once to the further relations % = V ft/ = 1,2, 3). The matrix in question must then be symmetrical. If, on the other hand, we have a u = a^ = a^ = 9 there must be some coefficient a^ different from 0. Suppose, for example, # 12 * 0. Then the Delation a 12 Aa 21 = shows that neither k nor # 21 can be 0. The substitution of one in. the other of the relations a 12 =&& 21 and a^ Jea^ then gives Jc* = 1, or Jc = 1. The value 7c = 1 again leads to the condition that the matrix of the coefficients be symmetrical The value & = 1 gives a K = 0, and # y = <^, which would render the matrix skew symmetrical. The determinant of the transformation would on this supposition vanish (since every skew-symmetrical determinant of odd order vanishes), which is contrary to the hypothesis. The value Jc BBS 1 is therefore impossible. We have thus been led to the fol- lowing theorem: THEOREM 7. TJie necessary and sufficient condition that a reci- procity in a plane le a polarity is that the matrix of its coefficients le symmetrical. If the coordinate system is chosen so that the point which corre- sponds to p =s 1 in Equation (2) is (1, 0, 0), it is clear that we must have a zl a 12 and # 81 = fl ia . If the line corresponding doubly to (1, 0, 0) does not pass through it, the coordinates [1, 0, 0] may be assigned to this line. The equations of the correlation thus assume the form (3) and Equation (2) reduces to (4) = 0. The roots, other than 1, of this equation clearly correspond to points on [1, 0, 0], Choosing one of these points (Proposition K 2 ) as (0, 0, 1), we have either # 28 = a S2 , which would lead to a polarity, or # 88 = 0. 97] DOUBLE PAIES OF A COEBELATION 281 In the latter case it is evident that (4) has a double root if # 32 = a 23 , but that otherwise it has two distinct roots. Therefore a correlation in which (1, 0, 0) and [1, 0, 0] correspond doubly, and which is not a polarity, may be reduced to one of the three forms : (a = 0, 1 = 0) K= - The squares of these correlations are collineations of Types /, //, IV respectively. If the line doubly corresponding to (1, 0, 0) does pass through it, the coordinates [0, 1, 0] may be assigned to this line, and the equa- tions of the correlation become Equation (2) at the same time reduces to (i-p)*-o, and the square of the correlation is always of Type III. There are thus five types of correlations, the polarity and those whose squares are collineations of Types /, II, III, IV. EXERCISES * 1. The points which lie upon the lines to which they correspond in a cor- relation form a conic section C 2 , and the lines which lie upon the points to which they correspond are the tangents to a conic K 2 . How are C 2 and E? related, in each of the five types of correlations, to one another and to the doubly corresponding elements ? * On the theory of correlations see Seydewitz, Archiv der Mathematik, 1st series, Vol. VIII (1846), p. 32 ; and Schr5ter, Journal fur die reine und angewandte Mathe- matik, Vol. LXXVII (1874), p. 105. 282 TWO-DIMENSIONAL PROJECTIVITIES [CHAP. X 2. If a line a does not lie upon the point A' to which it corresponds in a ' correlation, there is a projectivity between the points of a and tho points in which their corresponding lines meet a. In the case of a polarity this pro- jectivity is always an involution. In any other correlation the linos upon which this projectivity is involutoric all pass through a unique fixed point 0. The line o having the dual property corresponds doubly to 0. The doublo points of the involutions on the lines through are on the conic ( '-, and th< k double lines of the involutions on the points of /C 2 are tangent to A" 2 . and o are polar with respect to C 2 and K z . If a correlation determines involutions on three nonconcurrent lines, it is a polarity. 3. The lines of K 2 through a point P of C 2 are the line which is transformed into P and the line into which P is transformed by the given correlation. 4. In a polarity C 2 and K* are the same conic. 5. A necessary and sufficient condition that a collineation be the product of two reflections is the existence of a correlation which is left invariant by the collineation.* 98. Fundamental conic of a polarity in a plane. We have just seen that a polarity in a plane is given by the equations (1) DEFINITION. Two homologous elements of a polarity in a plane are called pole and polar, the point being the pole of the line and the line being the polar of the point. If two points are so situated that one is on the polar of the other, they are said to be conjugate. The condition that two points in a plane of a polarity he conju- gate is readily derived. In fact, if two points P = (M V x z , $ a ) and p* '=(#/, # a ', x) are conjugate, the condition sought is simply that the point P ! shall be on the line p f = \u[, u[, M 8 '], the polar of J\\ i.e. u[x{ + u&l + u&l = 0. Substituting for u[, ut, %[ their values in terms of x v x^ x s from (1), we obtain the desired condition, via. : (2) a^xl + As was to be expected, this condition is symmetrical in the coordi- nates of the two points P and P f . By placing x{ = as, we obtain the * This is a special case of a theorem of Dunham Jackson, Transactions of the American Mathematical Society, Vol. X (1909), p. 479, POLAR SYSTEM 283 condition that the point /' be self -conjugate, ie. that it be on its polar. We thus uUuiu the result; THKOUBM 8. 77/0 sdfaonjugttte print* of the polarity (1) are on the- MUM whoso equation is ft, r<f + <typ* + a^l + 2 a^v,, + 2 fl w avcfr + 2 a 2g aj a a? 8 = ; ( 9 conversely, every point of this conic is sdf -conjugate. This conic is called the fundamental conic of the polarity. All of it points may be improper, but it can never degenerate, for, if so, the determinant v | would have to vanish (cf. Ex., p. 187). By duality we obtain THEOREM 8 r . The self-c,o>iju<jate lines of tlie polarity (1) are lines of the, conic (4) A^ 4- A^l 4- J S X + 2' A^u^ 4- 2 A^n^ 4- 2 ^ 23 w 3 ^ 3 = ; <mrf, emwrMly, every line of this conic is self-conjugate. Every point A' of the conic (3) corresponds in the polarity (1) to the tangent to (3) at Jif. For if not, a point A of (3) would be polar to a line a through A and meeting (3) also in a point B> B would then be polar to a line I through B, and hence the line a^AB would, by the definition of a polarity, be polar to ab = B. This would require that a correspond both to A and to B. If now we recall that the polar system of a conic constitutes a polarity (Theorem 18, Cor., Chap. V) in which all the points and lines of the conic, and only these, are self-conjugate, it follows from the above that every polarity is given by the polar system of its fundamental conic. This and other results following immediately from it are contained in the following theorem : THEOREM 9. Mrery polarity w the polar &y$t&ni of a conic, the fwudaviental conic of the polarity. The self-conjugate points are the points and the self -conjugate lines are the tangents of this conic. Jtfvcry yoU and polar pair are pole and polar with respect to the fundamental conin. This establishes that Equation (4) represents the same conic as Equation (3). The last theorem may be utilized to develop the ana- lytic expressions for poles and polars, and tangents to a conic. This we take up in the next section. 284 TWO-DIMENSIONAL PEOJECTIVITIES |CHAI>. x 99. Poles and polars with respect to a conic. Tangents. We have seen that the most general equation of a conic in point coor- dinates may be written (1) a n x%+ & 22 # 2 2 + ^as&'s + 2 a u x l x 2 + 2 a 1 ^x 1 x s + 2 ^ 23 ^ 2 ^ 3 = 0. The result of the preceding section shows that the equation of the same conic in line coordinates is (2) 4X + A^ul + A^ul + 2 A^u, + 2 A^u, + 2 A n u t u s = 0, where A, is the cofactor of a,, in the determinant This result may also be stated as follows : THEOREM 10. The necessary and sufficient condition that the line w 1 a? 1 +^ a a; 3 + W 8 a? a =s le tangent to the conic (1) is that JSyitation (2) "be satisfied. COKOLLAKY. This condition may also le written in the form u, u n u 9 = 0. Equation (2) of the preceding section expresses the condition that the points (x v # a , # 3 ) and (#/, x 9 ojj) be conjugate with respect to the conic (1). If in this equation (a?/, x, &) be supposed given, while (x v # 2 , # 3 ) is regarded as variable, this condition is satisfied by all the points of the polar of (x[, x^, x^) with respect to the conic and by no others. It is therefore the equation of this polar. When arranged according to the variable coordinates x it it becomes (3) (*!!/+ ajX+ *vf&*i + (*vPi + * M atf + ^<)^2 , ; while if we arrange it according to the coordinates #?/, it becomes (4) (a lA + a lz x, + a lA ) x[ + (a, A + ^ 2 A + a^ < == 0. Now it is readily verified that the latter of these equations may be derived from the equation (1) of the conic by applying to the left-hand member of this equation the polar operator oi),ioo] VARIOUS DEFINITIONS OF CONICS 285 ( 89) and dividing the resulting equation by 2. Furthermore, if we define the symbols ~~ > -^ > -^ to be the result of substituting dx l dx^ 0a? 8 (^/, 2 ', # 3 ') for (ajj, a? a , a? 8 ) in the expressions -^-> ~^> -^- (/ being any </i#.| ve/t'g </cH2g polynomial in x v % 2 , X Q ), it is readily seen that Equation (3) is equivalent to where now/ is the left-hand member of (1). This leads to the following theorem : THEOREM 11. Iff^Qis the equation of a conic in homogeneous point coordinates, the equation of the polar of any point (%{, x 9 x$ is given ly either of the equations = ( ) or Xif + x 8x 8 *dx[ z d% o% If the point (%[, x^ x) is a point on the conic, either of these equa- tions represents the tangent to the conic f = at this point. 100. Various definitions of conies. The definition of a (point) conic as the locus of the intersections of homologous lines of two protective flat pencils in the same plane was first given by Steiner in 1832 and used about the same time by Ghasles. The considerations of the preceding sections at once suggest two other methods of defi- nition, one synthetic, the other analytic. The former begins by the synthetic definition of a polarity (cf. p. 263), and then defines a point conic as the set of all self-conjugate points of a polarity, and a line conic as the set of all self -con jugate lines of a polarity. This defini- tion was first given by wn Staudt in 1847. From it he derived the fundamental properties of conies and showed easily that his definition is equivalent to Steiner's. The analytic method is to define a (point) conic as the set of all points satisfying any equation of the second degree, homogeneous in three variables % v x z , x 3 . This definition (at least in its nonhomogeneous form) dates back to Descartes and Fermat (1637) and the introduction of the notions of analytic geometry. 286 TWO-DIMENSIONAL PBOJECTIVITIBB [CHAP. X The oldest definition of conies is due to the ancient Greek geometers, who defined a conic as the plane section of a circular cone. This definition involves metric ideas and hence does not concern us at this point. We will return to it later. It is of interest to note in passing, however, that from this definition Apollonius (about 200 A.D.) derived a theorem equivalent to the one that the equation of a conic in point coordinates is of the second degree. The reader will find it a valuable exercise to derive for himself the fundamental properties of polarities synthetically, and thence to develop the theory of conies from von Staudt's definition, at least so far as to show that his definition is equivalent to Steiner's. It may be noted that von Staudt's definition has the advantage over Steiner's of including, without reference to Proposition K 2 , conies consisting entirely of improper points (since there exist polarities which have no proper self-conjugate points). The reader may in this connection refer to the original work of von Staudt, Die Geometric der Lage, Niirnberg (1847)'; or to the textbook of Enriques, Vorlesungen liber projective G-eometrie, Leipzig (1903). EXERCISES 1. Derive the condition of Theorem 10 directly by imposing tho condition that the quadratic which determines the intersections of the given line with the conic shall have equal roots. What is the dual of this theorem ? 2. Verify analytically the fundamental properties of poles and polars with respect to a conic (Theorems 13-18, Chap. V). 3. State the dual of Theorem 11. 4. Show how to construct the correlation between a plane of points and a plane 'of lines, having given the homologous pairs A, of\ B, &'; C, cf\ 7>, d'. 5. Show that a correlation between two planes is uniquely determined if two pencils of points in one plane are made projective respectively with two pencils of lines in the other, provided that in this projectivity the point of intersection of the axes of the two pencils of points corresponds to the lino joining the two centers of the pencils of lines. 6. Show that in our system of homogeneous point and line coSrdinaten the pairs of points and lines with the same coordinates are poles and polars with respect to the conic x * + #| + xj = 0. 7. On a general line of a plane in which a polarity has boon defined the pairs of conjugate points form an involution the double points of which are the (proper or improper) points of intersection of the line -with the funda- mental conic of the polarity, 8. A polarity in a plane is completely defined if a self-polar triangle is given together with one pole and polar pair of which the point is not on a side nor the line on a vertex of the triangle. f 100, 101] PAIRS OF CONICS 287 9. Prove Theorem tt analytically. 10. G iven a simple plane pentagon, there exists a polarity in which to each vertex corresponds the opposite side. 11. The three points A', />", 6" on the sides BC, CA 9 AI)otQ, triangle that aro conjugate in a polarity to the vertices A> />, C respectively are collinear (cf. Ex. liJ, p. 125). 12. Show that a polarity is completely determined when the two involutions o conjugate points on two conjugate lines are given. 13. Construct the polarity determined by a self-polar triangle ABC and an involution of conjugate points on a line. 14. Construct the polarity determined by two pole and polar pairs A, a and />, // and one pair of conjugate points C, C'. 15. If a triangle RTU is self-polar with regard to a conic C 2 , and A is any point of G' 2 , there are three triangles having A as a vertex which are inscribed to C Y2 and circumscribed to STU (ISturm, Die Lehre von den geometrischen Verwandtschaften, Vol. I, p. 147). 101, Pairs of conies. If two polarities, i.e. two conies (proper or improper), are given, their product is a collineatiorx which leaves invariant any point or line which has the same polar or pole with regard to both conies. Moreover, any point or line which is not left invariant by this collineation must have different polars or poles with regard to the two conies. Hence the points and lines which have the same polars and poles with regard to two conies in the same plane form one of the five invariant figures of a nonidentical collineation. Type I. If the common self-polar figure of the two conies is of Type /, it is a self-polar triangle for both conies. Since any two conies are protectively equivalent (Theorem 9, Chap. VIII), the coordinate system may be so chosen that the equation of one of the conies, Jt a , is (1) itf-asJ + atf^O. With regard to this conic the triangle (0, 0, 1), (0, 1, 0), (1, 0, 0) is self-polar. The general equation of a conic with respect to which this triangle is self-polar is clearly (2) a ^ a^l + # A 2 = 0. An equation of the form (2) may therefore be taken as the equation of the other conic, # 2 , provided (1) and (2) have no other self-polar elements than the fxindamental triangle. Consider the set of conies (8) a^?- &A 2 -h o^ + XOaf- *?+) = 0. 288 TWO-DIMENSIONAL PRO JBCTIVITIBS [CHAP. X The coordinates of any point which satisfy (1) and (2) also satisfy (3), Hence all conies (3) pass through the points common to * and !> >a . For the value X = a s> (3) gives the pair of lines (4) (a, - a,) x* - fa - a,) v* = 0, which intersect in (0, 0, 1). The points of intersection of these lines with (1) are common to all the conies (3). The lines (4) are distinct, unless a^ a^ or <x 2 = <V But if ] = ,,, any point (xf, 0, x) on the line # 2 =0 has the polar x^ + x^ = both with regard to (1) and with regard to (2). The self-polar figure is therefore of Type IV. In order that this figure be of Type 7, the three numbers a v a z> a z must all be distinct. If this condition is satisfied, the lines (4) meet the conies (3) in four distinct points. FIG. 100 The actual construction of the points is now a problem of the second degree. We have thus established (fig. 100) THEOREM 12. If two conies have a common self -polar triangle (and no oilier common self -polar pair of point and line), they intersect in four distinct points (proper or improper). Any two conies of the pencil determined "by these points ham the same self -polar triangle. Dually, two such conies have four common tangents, and any two wi] PAIRS OF CONICS 289 B=(010) eowks of tlie rttnye daterwiucd "by those common taiigents have the same self-j>ol(tr tri<mgk. CoKoiJAuy. Any pencil of comas of Type I can be represented by * (5) (^- afl + X (-) =0, 4& the four common points being in this case (1,1, 1), (1, 1, 1), (1, 1, 1), 1,1,1). //. When the common self-polar figure is of Type //, one of the points lies on its polar, and therefore, this polar is a tangent to eacli of the conies A*, jB fl . Since two tangents cannot intersect in a point of contact, the two lines of the self-polar figure are not both tan- gents. Hence the point JB p IGf 101 of the self-polar figure which is on only one of the lines is the pole of the line I of the figure which is on only one of the points (fig. 101), and the line a on the two points is tangent to both conies at the point A which is on the two lines, Choose a system of coordinates with ^ = (1,0, 0), a=[0, 0, 1], #= (0, 1, 0), and i= [0, 1, 0]. The equation of any conic being 2 *=* > the condition that A be on the conic is ^=0 ; that & then be tan- gent is Z> 8 = 0; that I then be the polar of J3 is ^=0. Hence the general equation of a conic with the given self-polar figure is (6) <ytf + <V* + 2 * Equation (6) is typical for a pencil of conies of Type I, and Theorem 12 is a sort of converse to the developments of 47, Chap, V. The reader will note that if the problem of finding the points of intersection of two conies is set up directly, it is of the fourth degree, but that it is here reduced to a problem of the third degree (the determination of a common self-polar triangle) followed by two quad- ratic constructions. This corresponds to the well-known solution of the general biquadratic equation (cf. Fine, College Algebra, p. 486). For a further discussion of the analytic geometry of pencils of conies, cf . Clebsoh-Lindemann, Vorlesungen iiber Geometrie, 2d ed., Vol. I, Part I (1906), pp. 212 ff. 290 TWO-DIMENSIONAL PROJBOTIVITIES LCHAP. x Since any two conies are projectively equivalent, A 1 may be chosen to be (7) +;+ 2 3^=0. The equation of B* then has the form (6), with the condition that the two conies have no other common self-polar elements. Since the figure in which a is polar to A and b to B can only reduce to Types IV or F, we must determine under what conditions each point on a or each point on I has the same polar with regard to (6) and (7). The polar of (x[, x, A- 8 ') with regard to (6) is given by 1^3= 0- Hence the first case can arise only if & 2 = & 2 ; and the second only if a = 6 . > s Introducing the condition that a z , 8 , 5 2 are all distinct, it is then clear that the set of conies af* + a,x* + 2 l&Xi + X (xj + ^ + 2 x A ) = contains a line pair for X = # 2 , viz. the lines Hence the conies have in common the points of intersection with (7) of the line (o,-a a ) 0,+ 2(5,- ^=0. This gives THEOREM 13. If two conies have a common self -polar figure of Type II, tliey have three points in common and a common tangent at one of them. Diially, tliey have three common tangents and a common point of contact on one of the tangents. The two conies determine a, pencil and also a range of conies of Type II. COROLLARY. Any pencil of conies of Type II may le represented ly the equation x% # 8 2 + \x^ = 0. The conies of this pencil all pass through the points (0, 1, 1), (0, 1, 1), (1, 0, 0) and are tangent to # 3 = 0. Type III. When the common self-polar figure is of Type ///, the two conies evidently have a common tangent and a common point of contact, and only one of each. Let the common tangent be # 8 =s 0, its point of contact be (1, 0, 0), and let A 2 be given by (8) ^+ PAIRS OF CWICS 291 The general equation of a conic tangent to # 3 = at (1, 0, 0) is ( 8) <Vtt? + <v + 2 MA + 2 MA = 0, with regard to which the polar of any point (#/, &/, 0) on # 3 = is given l)y ( 1 0) This will be identical with the polar of (a x^ 0) with regard to A* for all values of as/, ai if &,,= 2 and \*= 0. Since (1, 0, 0) only is to have the same polar with regard to both conies, we impose at least one of the conditions & 2 =tf 2 , ^=0. The line (10) will now be identical with the polar of (8) for any point (x[, #/, 0) satisfying the condition This quadratic equation must have only one root if the self-polar figure is to be of Type ///. This requires & 2 = a a , and as J 2 , a 2 cannot both l)e unless (9) degenerates, the equation of B* can be taken as (11) xl + 2 x^ + off + 2 MA = 0, (B 1 *= 0). A~(lQO) a=[ooi] The conies (8) and (11) now evidently have in common the points of intci'HGction of (8) with the line pair 8 a; 8 fl + 2^8^8=0, and no other points. Since $*% = is a tangent, this gives two common points. If the second common point is taken to be (0, 0, 1), the set of conies which have in com- mon the points (0, 0, 1) and (1, 0, 0) = A and the tangent a at A 9 and no other points, may be written (fig. 102) (1 2) <* + 2 fa + \% 2 x s = 0. THEOREM 14. If two conies have a common self-polar figure of Type III t thfiy have two points in common and a common tangent at one of them, and one other common tangent. They determine a pencil and a range of conies of Type III. 292 TWO-DIMENSIONAL PROJECTIVITIES [OHAP.X COROLLARY. A pencil of conies of Type III can "be represented ly the equation x%+ 2 #3^ + \a? 2 # 3 = 0. Type IV. When the common self-polar figure is of Type 7F, let the line of fixed points be X B = and its pole be (0, 0, 1). The coordinates being chosen as they were for Type /, the conic A* has the equation X^ - XQ + #?g = U J and any other conic having in common with A* the self-polar tri- angle (1, 0, 0), (0, 1, 0), (0, 0, 1) has an equation of the form 2== 0. The condition that every point on # 8 = shall have the same polar with regard to this conic as with regard to A* is a l = a y Hence B may be written Any conic of this form has the same tangents as A 2 at the points (1, 1, 0) and (1, 1, 0) (fig. 103). Hence, if X is a variable parameter, the last equation represents a pencil of conies of Type IV according to the classification previously made. THEOREM 1 5. If two conies l la w a common self-polar figure of Type IV, they have Fio 103 two points in common and the tangents at these points, They determine a pencil (which is also a range) of conies of Type IV. COKOLLARY. A pencil of conies of Type IV may le represented ly the equation x* and also by the equation Type V. When the common self-polar figure is of Type F, let the point of fixed lines be (1, 0, 0) and the line of fixed points be # 8 = 0. As in Type ///, let J? be given by (8) ^+2^3=0. We have seen, in the discussion of that type, that all points of x^ have the same polars with respect to (8) and (9), if in (9) we have 101] PAIES OF CONICS 293 Z> a ss a% and l^ 0. Hence, if A* and J? 2 are to have a common self- polar figure of Type V, the equation of & must have the form (13) a.(x*+2x A )+a^*~Q. Prom the form of equations (8) and (13) it is evident that the conies have in common only the point (1, 0, 0) and the tangent X B = 0, and that every point on # 8 = has the same polar with ro- o,= [o0lj A = (\wo) spect to both conies (fig. 104). Hence they determine a pencil of Type V. THEOREM 16. If two conies have a common self -polar figure of Type V, they have a lineal element (and no other elements) in com- mon and determine a pencil (which is also a range) of conies of Type V according to the classification already given. COROLLAHY. A pencil of conies of Type V can le represented ly the equation 1 * As an immediate consequence of the corollaries of Theorems 12-16 we have THEOREM 17. Any pencil of conies may le written in the form where f~ and #= are the equations of two conies (degenerate or not) of the pencil. EXERCISES 1. Prove analytically that the polars of a point P with, respect to the conies of a pencil all pass through a point 'Q. The points P and Q are double points of the involution determined by the conies of the pencil on the line PQ. Give a linear construction for Q (cf. Ex. 3, p. 13 C). The correspondence obtained by letting every point P correspond to the associated point Q is a < < quadratic birational transformation. ' ' Determine the equations representing this transformation. The point Q, which is conjugate to P with regard to all the conies of the pencil, is called the conjugate of P with respect to the pencil. The locus of the conjugates of the points of a line with regard to a pencil of conies is a conic (cf. Ex. 81, p. 140). 2. One an<J only one conic passes through four given points and has two given points as conjugate points, provided the two given points are not con- jugate with respect to all the conies of the pencil determined by the given set of four. Show how to construct this conic. 294 TWO-DIMENSIONAL PEOJECTIVITIES [CHAP.X 3. One conic in general, or a pencil of conies in a special case, passes through three given points and has two given pairs of points as conjugate points. Give the construction. 4. One conic in general, or a pencil of conies in a special case, passes through two given points and has three pairs of given points as conjugate points ; or passes through a given point and has four pairs of given points as conjugate points ; or has five given pairs of conjugate points. Give the cor- responding constructions for each case. 102. Problems of the third and fourth degrees.* The problem of constructing the points of intersection of two cdhics in the same plane is, in general, of the fourth degree according to the classifi- cation of geometric problems described in 83. Indeed, if one of the coordinates be eliminated between the equations of two conies, the resulting equation is, in general, an irreducible equation of the fourth degree. Moreover, a little consideration will show that any equation of the fourth degree may be obtained in this way. It results that every problem of the fourth degree in a plane may be reduced to the problem of constructing the common points (or by duality the common tangents) of two conies. Further, the prob- lem of finding the remaining intersections of two conies in a plane of which one point of intersection is given 1 , is readily seen to be of the third degree, in general; and any problem of this degree can be reduced to that of finding the remaining intersections of two conies of which one point of intersection is known. It follows that any problem of the third or fourth degree in a plane may be reduced to that of finding the common elements of two conies in the plane, f A problem of the fourth (or third) degree cannot therefore be solved by the methods sufficient for the solution of problems of the first and second degrees (straight edge and compass). | In the case of problems of the second degree we have seen that any such prob- lem could be solved by linear constructions if the intersections of * In this section wo have made use of Amodeo, Lezioni di Ooomctria Projettiva, pp. 40(5, 437. Some of the exercises are taken from the same book, pp. 448-451. 1 Moreover, we have soon (p, 28!), footnote) that any problem of the fourth degree may be reduced to one of the third degree, followed by two of the floc.ond degree. | "With the usual representation of the ordinary real geometry \vo should require an instrument to draw conies. H)2] TI1IJID AND FOURTH DEGKEE PROBLEMS 295 every line in the plane with a fixed conic in that plane were assumed known. Similarly, any problem of the fourth (or third) degree can he solved by linear and quadratic constructions if the intersections of every conic in the plane with a fixed conic in this plane are assumed known. This follows readily from the fact that any conic in the plane can "be transformed by linear constructions into the fixed conic. A problem of the third or fourth degree in a plane will then, in the future, be considered solved if it has been reduced to the finding of the intersections of two conies, combined with any linear or quadratic constructions. As a typical problem of the third degree, for example, we give the following: To find the double points of a nonpcrspective collineation in a plane 'which is* determined ly four pairs of homologous points. Solution. When four pairs of homologous elements are given, we can construct linearly the point or line homologous with any given point or line in the plane. Let the collineation be represented by II, and lot A be any point of the plane which is not on an invariant line. Let U(A)=A r and II (A 1 )** A". The points A, A 1 , A" are then not collinear. The pencil of lines at A is protective with the pencil at A 1 , and these two projective pencils generate a conic C 2 which passes through all the double points of II, and which is tangent at A' to the line A r A" (fig. 105). The conic <7 2 is transformed by the collineation II into a conic C? generated by the pro- jective pencils of lines at A f and A". FlQ " 10 6\ a also passes through A! and is tangent at this point- to the line A A 1 . The double points of II are also points of C*. The point A f is not a double point of II by hypothesis. It is evident, however, that every other point common to the two conies C 2 and C* is a double point. If (7 s and C? intersect again in three distinct points Z, H> N 9 the latter form a triangle and the collineation is of Type /. If <7 2 and C intersect in a point N, distinct from A f , and are tangent to each other at a third point L=zM, the collineation has M, N for double points 296 TWO-DIMENSIOKAL PEOJECTIVITIES [CHAP.X and the line MN and the common tangent at M for double lines (fig, 106); it is then of Type //. If, finally, the two couics have contact of the second order at a point L = M=* N, distinct from A r , the collineation has the single double line which is tangent to the conies at this point, and is of Type III (fig. 107). , 106 tfw. 107 EXERCISES 1. Give a discussion of the problem above without making at the outset the hypothesis that the collineation is nonperspective. 2. Construct the double pairs of a correlation in the piano, which is not a polarity. 3. Given two polarities in a plane, construct their common polo and polar pairs. 4. On a line tangent to a conic at a point A is given an involution I, and from any pair of conjugates P, P f of I are drawn the second tangents p, p' to the conic, their points of contact being Q, Qf respectively. Show that the IOCUH of the point pp' is a line, /, passing through the conjugate, A', of A in the invo- lution I ; and that the line QQ' passes through the pole of I with respect to the conic. 5. Construct the conic which is tangent at two points to a given conic and which passes through three given points. Dualize. 6. The lines joining pairs of homologous points of a noninvolutoric pro- jectivity on a conic A* are tangent to a second conic B* which is tangent to A 2 at two points, or which hyperosculates A*. 7. A pencil of conies of Type // is determined by three points A, B, C and a line c through C. What is the locus of the points of contact of tho conies of the pencil with the tangents drawn from a givn point P of <?? 8. Construct the conies which pass through a given point P and which are tangent at two points to each of two given conies. 9. If /= 0, g s= 0, h = are the equations of thrcd couicH in a piano not belonging to the same pencil, the system of conies given by the equation JOiiJ THIRD AND FOURTH DEGREE PROBLEMS 297 X, ju,, v being variable parameters, is called a bundle of conies. Through every point of the plane passes a pencil of conies belonging to this bundle ; through any two distinct points passes in general one and only one conic of the bundle. If the conies /, r/, h have a point in common, this point is common to all the conies of the bundle. Give a nonalgebraic definition o a bundle of conies. 10. Tho set of all conies in u plane passing through the vertices of a triangle form a bundle. Tf the equations of the sides of this triangle are I = 0, m = 0, n = 0, show that the bundle may be represented by the equation Xwn + jjinl + vim = 0. What aro the degenerate conies oi this bundle ? * 11. The not of all conies in a plane which have a given triangle as a self- polar triangle forms a bundle. If the equations of the sides of this triangle are / = 0, m 0, n = 0, show that the bundle may be represented by the equation A/ 2 4- /im a + w* = 0. What aro the degenerate conies of this bundle? 12. The conies of the bundle described in Ex. 11 which pass through a general point P of the plane pass through the other three vertices of the quadrangle, of which one vertex is P and of which the given triangle is the diagonal triangle. What happens when P is on a side of the given triangle ? Dualize. 13. The reflections whose centers and axes are the vertices and opposite aides of a triangle form a commutative group. Any point of the plane not on a side of the triangle is transformed by the operations of this group into the other three vertices of a complete quadrangle of which the given triangle is the diagonal triangle. If this triangle is taken as the reference triangle, what are the equations of transformation ? What conies are transformed into themselves by the group, and how is it associated with the quadrangle- quadrilateral configuration ? 14. The necessary and sufficient condition that two reflections be com- mutative Is that the center of each shall be on the axis of the other. 15. The invariant figure of a collineation may be regarded as composed of two lineal elements, the five types corresponding to various special relations between the two lineal elements. 16. A correlation which transforms a lineal element Aa into a lineal clement 7#; and also transforms Bl into -la is a polarity. 17. How many collineations and correlations are in the group generated by the. reflections whoso centers and axes are the vertices and opposite sides of a triangle and a polarity with regard to which the triangle is self -polar ? * In connection with this and the two following exercises, cf . Castelnuovo, Lessioni di Geornetria Analitica e Projettiva, Vol. I, p. 395. CHAPTER XI* FAMILIES OF LINES 103- The regulus. The following theorem, on which depends the existence of the figures to be studied in this chapter, IKS logically equivalent (in the presence of Assump- tions A and E) to Assumption P. 1 1 might have been used to replace that assumption. THEOREM 1. If l v 1 S9 1 9 are three, mutally skew lines, and if m v w 2 , w 8 , w 4 are four lines each of which meets each of the lines l v /, 7 3 , then any line l^ which meets three of the lines m v m^ m s , m^ also meets the fourth. Proof. The four planes l$n v Ijn^ Z x m 3 , lp\ of the pencil with axis l l are perspective through the pencil of points on 1 3 with the four planes l z m v / a w fi , fy j h> ^2 m 4 ^ ^ P enc il with axis / a (fig. 108). For, by hypothesis, the lines of intersection m v w 2 , m^ m^ of the pairs of homologous planes all meet / 8 . The set of four points in which the four planes of the pencil on ^ meet Z 4 is therefore projective with the set of four points in which the four planes of the pencil on 2 meet Z 4 . But l meets three of the pairs of homologous planes in points of their lines of intersection, since, by hypothesis, it meets three of the lines m v m 2 , w 3 , w 4 . Hence in the projectivity on l there are three invariant points, and hence (Assumption P) every point is invariant. Hence l meets the remain- ing line of the set m v m^ m v m. * AH the developments of this chapter are on the basis of Assumptions A, E, P, HO. But see the exercise oa page 201. 208 108 SW] THE EEG-ULUS 299 DEFINITION. If l v Z 2 , l s are three lines no two of which are in the Banie plane, the set of all lines which meet each of the three given lines is called a regulus. The lines l v ,, 1% are called directrices of this regulus. It is clear that no two lines of a regains can intersect, for other- wise two of the directrices would lie in a plane. The next theorem follows at once from the last. THEOREM 2. If l v / 2 , Z 8 are three lines of a regulus of which m v v/6 a , w 8 are directrices, m v m 2 , m 3 are lines of the reyidus of which l v l, / 8 arc directrices. It follows that any three lines no two of which lie in a plane are directrices of one and only one regulus and are lines of one and only one regulus. DEFINITION. Two reguli which are such that every line of one meets all the lines of the other are said to be conjugate. The lines of a regulus are called its generators or rulers ; the lines of a conjugate regulus are called the directrices of the given regulus. THEOREM 3, Every regulus has one and only one conjugate regulus. This follows immediately from the preceding. Also from the proof of Theorem 1 we have THEOREM 4. Tlie correspondence THEOREM 4'. The correspond- established ly the lines of a regu- ence established "by the lines of a lux between the points of two lines regulus between the planes on any of its conjugate regulus is projec- two lines of its conjugate regulus tivc. is projectile. THEOR*^ 5. The set of all lines THEOREM 5'. The set of all lines of joining pairs of homologous points intersection of pairs of homologous of two projectioe pencils of points planes of two projectile pencils of on #7iYW lines is a regulus. planes on skew lines is a regulus. Proof. We may confine ourselves to the proof of the theorem on the left. By Theorem 6, Chap. Ill, the two pencils of points are perspective through a pencil of planes. Every line joining a pair of homologous points of these two pencils, therefore, meets the axis of the pencil of planes. Hence all these lines meet three (necessarily skew) lines, namely the axes of the two pencils of points and of the pencil of planes, and therefore satisfy the definition of a regulus. Moreover, every line which meets these three lines joins a pair of homologous points of the two pencils of points. 300 FAMILIES OF LINES LOUAP. XI THEOREM 6. If [p] are tlie lines of a rcyidus tt,nd q in u* dirwkrir, of tlie regulus, the pencil of points gfj?] 'is projcvtire with, tlie, pMwil of planes q [p]. Proof. Let g[ be any other directrix. By Theorem 4 the pencil of points q[p] is perspective with the pencil of points j'jjp]. But each of the points of this pencil lies on the corresponding piano <jj>. Hence the pencil of points < f [p] is also perspective with the pencil of planes q [p]. EXERCISES 1. Every point which is on a line of a rogulus is also on a line of its conjugate regains. 2. A plane which contains one line of a regnlus contains also a line of its conjugate regains. 3. Show that a regains is uniquely denned by two of its lines and throe of its points,* provided no two of the latter are coplanar with either of the given lines. 4. If four lines of a regains cut any line of the conjugate regains in points of a harmonic set, they are cut by every such line in points of a harmonic set. Hence give a construction for the harmonic conjugate of a line oi! a regains with respect to two other lines of the regains. 5. Two distinct reguli can have in common at most two distinct linen. 6. Show how to construct a regnlus having in common with a given regains one and but one ruler. 104. The polar system of a regulus. A plane meets every line of a regulus in a point, unless it contains a line of the regulus, in which case it meets all the other lines in points that are collinear. Since the regulus may be thought of as the lines of intersection of pairs of homologous planes of two projective axial pencils (Theorem fi'), the section by a plane consists of the points of intersection of pairs of homologous lines of two projective flat pencils. Hence the section of a regulus by a plane is a point conic, and the conjugate regulun has the same section. By duality the projection of a regulus and its conjugate from any point is a cone of planes. The last remark implies that a line conic is the "picture " in a piano of a regulus and its conjugate. For such a picture in clearly a plant) Hoctiou of the projection of the object depicted from the eye of an observer. Fig. 108 illustrates this fact. * By a point of a regains is meant any point on a line of the regains. iw] THE REGULUS 301 The section of a regiilus by a plane containing a line of the regu- lus is a degenerate conic of two lines. The plane section can never degenerate into two coincident lines because the lines of a regulus and its conjugate are distinct from each other. In like manner, the projection from a point on a line of the regulus is a degenerate cone of planes consisting of two pencils of planes whose axes are a ruler and a directrix of the regulus. DEFINITION. The class of all points on the lines of a regulus is called a surface of the second order or a guadric surface. The planes on the lines of the regulus are called the tangent planes of the sur- face or of the regulus. The point of intersection of the two lines of the regulus and its conjugate in a tangent plane is called the point of contact of the plane. The lines through the point of contact in a tangent plane are called tangent lines, and the point of contact of the plane is also the point of contact of any tangent line. The tangent lines at a point of a quadric surface include the lines of the two conjugate reguli through this point and all other lines through this point which meet the surface in no other point Any other line, of course, meets the surface in two or no points, since a plane through the line meets the surface in a conic. The tangent lines are, by duality, also the lines through each of which passes only cue tangent plane to the surface. THEOREM 7. TJw tangent planes at the points of a plane section of a quadric surface pass through a point and constitute a cone of planes. Dually, the points of contact of the cone of tangent planes through a point are coplanar and form a point conic. Proof. It will suffice to prove the latter of these two dual theorems. Let the vertex P of the cone of tangent planes be not a point of the surface. Consider three tangent planes through P, and their points of contact. The three lines from these points of contact to P are tan- gent lines o the surface and hence there is only one tangent plane through each of them. Hence they are lines of the cone of lines asso- ciated with the coue of tangent planes. Let TT be the plane through their points of contact. The section by TT of the cone of planes through P is therefore the conic determined by the three points of contact and the two tangent lines in which two of the tangent planes meet TT. The plane TT, however, meets the regulus in a conic of which the three points of contact are points. The two lines of intersection with 302 FAMILIES OF LINES LULU-. XI TT of two of the tangent planes througli P are tangents to this conic, because they cannot meet it more than one point each. The section of the surface and the section of the cone ot planes then have three points and the tangents through two of them in common. Hence these sections are identical, which proves the theorem when P IN not on the surface. If P is on the surface, the cone of planes degenerates into two lines of the regulus (or the pencils of planes on these lines), and the points of contact of these planes are all on the same two lines. Hence the theorem is true also in this case. DEFINITION. If a point P and a plane TT are so related to a regulus that all the tangent planes to the regulus at points of its section by TT pass through P (and hence all the points of contact of tangent planes through P are on TT), then P is called the pole of vr and TT the polar of P with respect to the regulus. COROLLARY. A tangent plane to a regulus is the polar of its point of contact. THEOREM 8. The polar of a point P not on a regidus contains all points P 1 such that the line PP 1 meets the siwface in two points which are harmonic conjugates with respect to P, PJ Proof. Consider a plane, a, through PP' and containing two lines a, Z> of the cone of tangent lines through P. This plane meets the surface in a conic <7 2 , to which the lines a f I are tangent. As the polar plane of P contains the points of contact of a and ft, its section by a is the polar of P with respect to <7 2 . Hence the theorem follows as a consequence of Theorem 13, Chap. V, THEOREM 9. The polar of a point of a plane TT with respect to a regulus meets TT in the 'polcvr line of this point with regard to the conic which is the section of the regulus ly TT. Proof. By Theorem 8 the line in which the polar plane meets TT has the characteristic property of the polar line with respect to a conic (Theorem 13, Chap. V). This argument applies equally well if the conic is degenerate. In this case the theorem reduces to the following COROLLARY. The tangent lines of a regulus at a point on it, are paired in an involution the double lines of which are the ruhr (tn,d directrix through that point. Each line of a pair contains the polar points of all the planes on the other line. w*J THE EEGULUS 303 THKOKBM 10. The polar s with regard to a regulus of the points of a line I are an axial pencil of planes projeetive with the pencil of points on L Proof. In case the given line is a line of the regulus this reduces to Theorem 6. In any other case consider two planes through I. In each plane the polars of the points of I determine a pencil of lines projeetive with the range on L Hence the polars must all meet the line joining the centers of these two pencils of lines, and, being per- spective with either of these pencils of lines, are projeetive with the range on /. DMFINITTON. A line V is polar to a line I if the polar planes of the points of I meet on V. A line is conjugate to I if it meets V. A point JP r is conjugate to a point P if it is on the polar of P. A line p is conjugate to P if it is on the polar of P. A plane TT' is conjugate to a plane TT if TT' is on the pole of ir. A line p is conjugate to TT if it is on the pole of TT. EXERCISES Polar points and planes with respect to a regulm are denoted "by corresponding capital Roman and small Greek letters. Conjugate elements of the same kind are denoted by the same letters wilh primes. 1. If TT is on It, then 7* is on g. 2. If Ms polar to /, then /is polar to /. 3. If one clement (point, line, or plane) is conjugate to a second element, then the second element is conjugate to the first. 4. If two lines intersect, their two polar lines intersect. 5. A ruler or a directrix of a regulus is polar to itself. A tangent line is polar to its harmonic conjugate with regard to the ruler and directrix through its point of contact. Any other line is skew to its polar. 6. The points of two polar lines are conjugate. 7. The pairs of conjugate points (or planes) on any line form an involu- tion the double points (planes) of which (if existent) are on the regulus. 8. The conjugate lines in a flat pencil of which neither the center nor the plane is on the regulus form an involution. 9. The line of intersection of two tangent planes is polar to the line joining the two points of contact. 10. A line of! the regulus which meets one of two polar lines meets the other. 11. Two one- or two-dimensional forms whose bases are not conjugate or polar are projeetive if conjugate elements correspond. 12. A line / is conjugate to I' if and only if some plane on / is polar to some point on /'. 304 FAMILIES OF LINES LO.IAP. xi 13. Show that there are two (proper or improper) lines ?, ,s nutting two given linos and conjugate to them "both. Show also that r is the polar of . 14. Jf , b, c are three generators of a regains and ', //, <' three of the con- jugate rcgulus, then the three diagonal lines joining the points (?;<?') and (Zi'e), (c'a) and (c'), (&') and (a'fi) meet in a point tf which is the pole of a plane containing the lines of intersec- tion of the pairs of tangent planes at the same vertices. 15. The six lines a, &, c, a', ft', c' of Ex. 14 determine the following trios of simple hexagons (bc'aVca') , (Aa'aeW) , (W/aa'cc") , (&c'aa'cft') , (bb'ac'ca') , (6a W/cc') . The points S determined by each trio of hexagons are collinear, and the two lines on which they lie are polar with regard to the quadric surface.* 16. The section of the figure of Ex. 14 by a plane leads to the Pascal and Brianchon theorems ; and, in like manner, Ex. 15 leads to the theorem that the 60 Pascal lines corresponding to the GO simple hexagons formed from 6 points of a conic meet by threes in 20 points which constitute 10 pairs of points conjugate with regard to the conic (cf. Ex. 10, p. 105. Protective conies. Consider two sections of a regains by planes which are not tangent to it. These two conies are both per- spective with any axial pencil of a pair of axial pencils which generate the regulus (cf. 76, Chap. VIII). The correspondence established between the conies by letting correspond pairs of points which lie on the same ruler is therefore projective. On the line of intersection, I, of the two planes, if it is not a tangent line, the two conies determine the same involution I of conjugate points. Hence, if one of them inter- sects this line in two points, they have these two points in common. If one is tangent, they have one common point and one common tangent. The projectivity between the two conies fully determines a projectivity between their planes in which the HBO I is transformed into itself. The involution I belongs to the projectivity thus deter- mined on I The converse of these statements leads to a theorem which -is exemplified in the familiar string models: THEOREM 11. The lines joining corresponding points of 9 two pro- jective conies in different planes form a reyul'U$ t provided GM two conies determine the same involution, T, of conjugate points on th.Q * Cf. Saimia, Lezioni di Geometria Projettiva (Naples, 1805), pp. 10SJ PKOJECTIVE CONICS 305 lin,e of intersection, I, of the two planes; and provided the collineation between the two pkmes determined ly the correspondence of the conies transforms I into itself ly a projectiviky to which I belongs (in par- ticular, 'if the conies meet in two points which are self -corresponding in> the. Proof. Let L be the pole with regard to one conic of the line of intersection, /, of the two planes (fig. 109). Let A and B be two points of this conic collinear with L and not on L The conic is gen- erated by the two pencils A [P] and B [P r ] where P and P f are con- jugates in the involution I on I (cf. Ex. 1, p. 137). Let A and B be the points homologous to A and B on the second conic, and let 2 be the point in which the second conic is_ met by the plane con- taining A, A> and the tangent at A\ and let B be the point in which the second conic is met by the plane of B, B, and the tangent at B. The line AB contains the pole of I with regard to the second conic because this line is projective with --4-??. Since the tangents to the first conic at A and B meet on Z, the complete quadrangle AABB has one diagonal point, the intersection of A A and BB, on I ; hence the 306 FAMILIES OF LINES l<!iui.xi opposite side of the diagonal triangle pauses through the polo of /, Hence it intersects AB in the pole of /. But the intersection of All with An is on this diagonal line. Hence All uieetH AB in the pole of /. Hence the pencils A[P] and B[P f "\ generate the second conic. Hence, denoting by a and 6 the lines A A and BB, the pencils of planes a [P] and I [P f ] are protective and generate a regulus of which the two conies are sections. The projectivity between the planes of the two conies established by this regulus transforms the line I into itself by a projectivity to which the involution I belongs and makes the point A correspond to A. The projectivity between two conies is fully determined by these conditions (cf. Theorem 12, Cor. 1, Chap. VIII). Hence the lines of the regulus constructed above join homologous points in the given projectivity. Q.E.D. It should be observed that if the two conies are tangent to I, the projectivity on I fully determines the projectivity between the two conies. For if a point P of I corresponds to a point Q of /, the unique tangent other than I through P to the first conic must correspond to the tangent to the second conic from Q. If the projectivity between the two conies is to generate a regulus, the projectivity on I must be parabolic with the double point at the point of contact of the conies with L For if another point D is a double point of the projectivity on I, the plane of the tangents other than I, through D to the two conies meets each conic in one and only one point, and, as these points are homologous, contains a straight line of the locus generated. As this plane contains only one point on either conic, it meets the locus in only one line, whereas a plane meeting a regulus in one line meets it also in another distinct line. Since the parabolic projectivity on I is fully determined by the double point and one pair of homologous points, the projectivity be- tween the two conies is fully determined by the correspondent of one point, not on Z, of the first conic. To show that a projectivity between the two conies which is para- bolic on I does generate a regulus, let A be any point of the first conic and A J its correspondent on the second (fig. 110). Let the plane of A 1 and the tangent at A meet the second conic in A n . Denote the common point of the two conies by , and consider the > 105 J PEOJECTIVE CONICS 307 FIG. 110 two conies as generated by the flat pencils at A and B and at A" and 11. The correspondence established between the two flat pencils at 7> by letting correspond lines joining B to homologous points of the two conies is perspective because the line I corresponds to itself. Hence there is a pencil of planes whose axis, Z>, passes through 7* and whose planes contain homologous pairs of lines of the flat pencils at 7A The correspondence established in like manner between the flat pencil at A and the flat pencil at ./I" may bo regarded as the product of the projectivity between the two planes, which car- ries the pencil at A to the pencil at A', followed by the projectivity between the pencils at A' and A n generated by the second conic. Both of these projectivities determine parabolic projectivities on I with B as inva- riant point. Hence their product determines on I either a parabolic projectivity with B as invariant point or the identity. This product transforms the tangent at A into the line A n A*. As these lines meet / in the same point, the projectivity determined on I is the identity. Hence corresponding lines of the projective pencils at A and A n meet on Z, and hence they determine a pencil of planes whose axis is a=AA lf . The axial pencils on a and & are projective and hence generate a regulus the lines of which, by construction, pass through homologous points of the two conies. We are therefore able to supplement Theorem 11 by the following COROLLARY 1. The lines joining corresponding points of two pro- jectwe conies in different planes form a regulus, if the two conies have a common tangent and point of contact and the projectivity determined between the two planes ly the projectivity of the conies transforms their common tangent into itself and has the common point of the two conies as its only fixed point. 808 FAMILIES OB 1 LINES [CHAP. XI Tlu. generation of a ivguiuH by projective ranges of points on skew linen may bo regarded UH a degenerate case of this theorem and eor- olluty, A further degenerate cane is stated in the first exercise, The proof of Theorem 11 given above is more complicated than it would lutve been if, under Proposition K 2 , we had made use of the points of intor- Hwtioii of the lino I with the two conies. But since the discussion of linear families of linen in the following section employs only proper elements and depends in part on this theorem, it seems more satisfactory to prove this theorem as we have done. It is of course evident that any theorem relating entirely to proper elements of space which is proved with the aid of Proposi- tion, K n can also be proved Ly an argument employing only proper elements. The latter form of proof is often much more difficult than the former, but it often yields more information as to the constructions related to the theorem. These results may be applied to the problem of passing a quadric surface through a given set of points in space. Proposition K 2 will be used in this discussion so as to allow the possibility that the two con- jugate reguli may be improper though intersecting in proper points. COROLLARY 2. If three planes a, fi, j meet in three lines a = /3% 6 = yor, e = a/3 and contain three conies A 2 , *, C 2 , of which Jf and C' 2 meet in two points P, P r of a, (? and A 2 meet in two points Q, Q r of I, and A 2 and B* meet in two points E, It' of e, then there is one and lut one qiiadric surface * containing the points of the three conic,*. Proof. let M be any point of <7 2 . The conic # 2 is projected from M by a cone which meets the plane a in a conic which intersects A* in two points, proper or improper or coincident, other than H and Ji 1 . Hence there are two lines m, m f , proper or improper or coincident, through M which meet both A* and B*. The projectivity determined between A* and JB* by either of these lines generates a regulus, or, in a special case, a cone of lines, the lines of which must pass through all points of <7 2 because they pass through P, P', Q r , Q, and M t all of which are points of <7 2 . The conjugate of such a regulus also contains a line through M which meets both A* and B*. Hence the lines m and m' determine conjugate reguli if they are distinct. If coincident they evidently de- termine a cone. The three conies being proper, the quadrio miittt con- tain proper points even though the lines rn, m f are improper. * In this corollary and in Theorem 12 the term quadric surface must be taken to include the points on a cone of lines as a special case. 1031 QUADEJU THROUGH NINE POINTS 309 9 If six points 1, 2, , 4, 5, 6 are given, no four of which are col- linear,* there evidently exist two planes, cc and /?, each containing three of the points and having none on their line of intersection. FIG. ill Assign the notation so that 1, 2, 3 are in #. A quadric surface which contains the six points must meet the two planes in two conies A 2 , If which meet the line #/3=c in a common point-pair or point of contact ; and every point-pair, proper or improper or coincident, of c determines such a pair of conies. Let us consider the problem .of determining the polar plane & of an arbitrary point on the line c. The polar lines of with regard to a pair of conies JL and B* meet c in the same point and hence determine o>. If no two of the points 1, 2, 3, 4, 5, 6 are collinear with 0, any line I in the plane a determines a unique conic A 2 with regard to which it is polar to 0, and which passes through 1, 2, 3. A* determines a unique conic 2? a which passes through 4, 5, 6 and meets c in the same points as -4 2 ; and with regard to this conic * The construction of a quadric surface through nine points by the method used in the text is given in Rohn'and Papperitz, Darstellende Geometric, Vol. II (Leipzig, 1806), 676, 677. 310 FAMILIES OF LINES [CHAP. XI has a polar line m. Thus there is established a one-to-one corre- spondence II between the lines of a and the lines of fi. This corre- spondence is a collineation. ' For consider a pencil of linen [I] in a. The conies J 2 determined by it form a pencil. Hence the point-pairs in which they meet c are an involution. Hence the conies 7> a deter- mined by the point-pairs form a pencil, and hence the lines [w] form a pencil. Since every line I meets its corresponding line m on c, the correspondence II is not only a collineation but is a pernpectivity, of which let the center be (7. Any two corresponding lines I and m are coplanar with C. Hence the polar planes of wiih regard to qiiadrics through 1, 2, 3, 4, 5, 6 are the planes on G. This was on the assumption that no two of the points 1, 2, 3, 4, 5, 6 are collinear with 0. If two are collinear with 0, every polar plane of must pass through the harmonic conjugate of with regard to them. This harmonic conjugate may be taken as the point C. Now if nine points are given, no four being in the same plane, the notation may be assigned so that the planes a 123, /? = 456, 7 = 780 are such that none of their lines of intersection a = 0% "b ~ yet, c = a/3 contains one of the nine points. Let be the point <xj3y (or a point on the line aft if a, ft, and 7 are in the same pencil). By the argu- ment above the polars of with regard to all quadrics through the six points in a and yS must meet in a point C. The polars of with regard to all quadrics through the six points in ft and 7 must simi- larly pass through a point A, and the polars with regard to all quad- rics through the six points in 7 and a must pass through a point /?. If A, B, and G are not collinear, the plane <o=xAnC must be the polar of with regard to any quadric through the nine points. The plane <a> meets a, ft, and 7 each in a line which must be polar to with regard to the section of any sucli quadric. But this determines three conies A* in a, B* in y8, and (7 2 in 7, which meet by pairs in three point-pairs on the lines a, 5, c. Hence if a, /3, 7 are not in the same pencil, it follows, by Corollary 2, that there is a unique quadric through the nine points. If a, /9, 7 have a line in common, the three conies A 2 , z , C* meet this line in the same point-pair. Consider a plane <r through which meets the conies A*, I? 2 , C 2 in three point- pairs. These point-pairs are separated harmonically by and the trace, $, oh <r of the plane G>. Hence they lie on a conic 7> a , which, with A 2 and P 2 , determines a unique quadric. The section of this ion, iixij LINEAH DEPENDENCE 0V LINES 311 quadrie by the plane 7 lias in common with # 2 two point-pairs and the polar pair 0, a. Hence the quadric has G 1 as its section by 7. Tu case A, 7?, and G are collinear, there is a pencil of planes CD which meet them. There is thus determined a family of quadrics which is called a pwml and is analogous to a pencil of conies. In case A, B, and G coincide, there is a bundle of possible planes a> and a quadric is determined for each one. This family of quadrics is called a bundle. Without inquiring at present under what conditions on the points 1, 2, , 9 these cases can arise, we may state the following theorem : THEOREM 1 2. Through nine points no four of which are collinear there JMMMS one quadric surface or a pencil of yuadrics or a bundle of qit,adric8. EXERCISES 1. The lines joining homologous points of a projective conic and straight lino form a rcguliiH, provided the line meets the conic and is not coplanar with it, and their point of intersection is self -corresponding. 2. State the duals of Theorems 11 and 12. 3. Show that two (proper or improper) 'conjugate reguli pass through two (ionics in different planes having two points (proper or improper or coincident) in common and through a point not in the plane of either conic. Two such conies and a point not in either plane thus detenninc one quadric surface. 4. Show how to construct a regulus passing through six given points and a given line. 106. Linear dependence of liaes. DEFINITION. If two lines are co- plauar, the lines of the flat pencil containing them both are said to be linearly dependent on them. If two lines are skew, the only lines linearly dependent on them are the two lines themselves. On three skew lines are linearly dependent the lines of the regulus, of which they are rulers. If l v l^ - , l n are any number of lines and m v m z ,--, m k are lines such that ^\ is linearly dependent on two or three of l v Z 2 , - - - , Z n , and w 2 is linearly dependent on two or three of l v 1 2 , , l n , m v and so on, m k being linearly dependent on two or three of l v l v , l n , m lt m< 2 , - , tn, kmml , then m k is said to be linearly dependent on l v l%> , l^ A set of n lines no one of which is linearly dependent on the n~l others is said to be linearly independent. As examples of these definitions there arise the following cases of linear dependence of lines on three linearly independent lines which may be regarded as degenerate cases of the regulus. (1) If lines a 312 FAMILIES OF LINES . [OIIAP.XI and I intersect in a point P, and a line c skew to both of them meets their plane in a point Q, then in the first place all lines of the pencil ctb are linearly dependent on a, &, and e ; since the line QP is in this pencil, all lines of the pencil determined by QP and c are in the sot. As these pencils have in common only the line QP and do not con- tain three mutually skew lines, the set contains no other linen. Hence in this case the lines linearly dependent on a, "b, c are the flat pencil ab and the flat pencil (c, QP). (2) If one of the lines, as a, meets both of the others, which, however, are skew to each other, the set of linearly dependent lines consists of the flat pencils ab and UG. This is the same as case (1). (3) If every two intersect but not all in the same point, the three lines are coplanar and all lines of their plane are linearly dependent on them. (4) If all three intersect in the same point and are not coplanar, th bundle of lines through their common point is linearly dependent on them. The case where all three are concurrent and coplanar does not arise because three such lines are not independent. This enumeration of cases may be summarized as follows : THEOREM 13. DEFINITION. The set of all lines linearly dependent on three linearly independent lines is either a regidus, or a "bundle, of lines, or a plane of lines, or two flat pencils having diffc-rcnt centers and planes hot a common line. The last three sets of lines are called degenerate reguli. DEFINITION. The set of all lines linearly dependent on four linearly independent lines is called a linear congruence. The set of all linen linearly dependent on five linearly independent lines is called a linear complex* 107. The linear congruence. Of the four lines a, ?;, 0, d upon which the lines of the congruence are linearly dependent, ft, v, d determine, as we have just seen, either a regulus, or two Hat pencils with different centers and planes but with one common lino, or a bundle of lines, or a plane of lines. The lines 7>, 0, d can of courso be replaced by any three which determine the same rcgulus or degen- erate regulus as &, c, d. * The terms congruence and complex are general terms to denote two- and three- parameter families of lines respectively, For example, all lines mooting a curve or all tangents to a surface form a complex, while all lines meeting two curves or all common tangents of two surfaces are a congruence. i<>7] THE LINEAll CONGRUENCE 313 So in case 5, c, d determine a nondegenerate regulus of which a is not a directrix, the congruence can be regarded as determined by four mutually skew lines. In case a is a directrix, the lines linearly de- pendent on a, ?>, c, d clearly include all tangent lines to the regulus bud, whose points of contact are on a. But as a is in a flat pencil with any tangent whose point of contact is on a and one of the rulers, the family of lines dependent on a, 5, c, d is the family de- pendent on ft, c, d and a tangent line which does not meet &, c, d. Hence in either case the congruence is determined by four skew lines. If one of the four skew lines meets the regulus determined by the other three in two distinct points, JP, Q, the two directrices p, q through these points meet all four lines. The line not in the regulus determines with the rulers through P and Q, two flat pencils of lines which join P to all the points of q, and Q to all the points of p. From this it is evident that all lines meeting loth p and q are linearly dtyenflvit on Ike given four. For if 1\ is any point on p, the line I\Q and the ruler through P l determine a flat pencil joining P to all the points of q; similarly, for any point of q. No other lines can be dependent on them, because if three lines of any regulus meet p and q, so do all the lines. If one of the four skew lines is tangent to the regulus determined by the other three in a point P, the family of dependent lines in- cludes the regulus and all lines of the flat pencil of tangents at P. Hence it includes the directrix p through P and hence all the tangent lines whose points of contact are on p. By Theorem 6 this family of lines can be described as the set of all lines on homologous pairs in a certain projectivity II between the points and planes of p. Any two lines in this set, if they intersect, determine a flat pencil of lines in the set. Any regulus determined by three skew lines I, m, n of the set determines a projectivity between the points and planes on p, but this projectivity sets up the same correspondence as II for the three points and planes determined by I, m, and n. Hence by the fundamental theorem (Theorem 17, Chap. IV) the projectivity deter- mined by the regulus Imn is the same as H, and all lines of the regulus are in the set. Hence, when one of four skew lines is tangent to the rqjulus of the other three, the family of depende?it lines consists of a regulus and all lines tangent to it at points of a directrix. The directrix is itself in the family. 314 FAMILIES OF LINES F. XI If no one of the four skew lines meets the regulas of the other three in a proper point, we have a case studied more fully below. In case 5, c } d determine two flat pencils with a common line, u. may meet the center A of one of the pencils. The linearly dependent lines, therefore, include the bundle whose center is A. The plane of the other flat pencil passes through A and contains throe noucon- current lines dependent on a, &, e, d. Hence the family of lines also includes all lines of this plane. The family of all lines through a point and all lines in a plane containing this point has evidently no further lines dependent on it. This is a degenerate case of a con- gruence. If a is in the plane of one of the flat pencils, we have, by duality, the case just considered. If a meets the common line of the two flat pencils in a point distinct from the centers, the two flat pencils may be regarded as determined by their common line d r and by lines V and c f , one from each pencil, not meeting a. Hence the family of lines includes those dependent on the regulus aW and its directrix d f . This case has already been seen to yield the family of all lines of the regulus aW and all lines tangent to it at points of d( Fia. 112 If a does not meet the common line, it meets the plants of tho two pencils in points and D. Call the centers of the pencils A and B (fig. 112). The first pencil consists of the lines dependent on AD and AB, the second of those dependent on An and BC. AH (ID is the line a, the family of lines is seen to consist of the lines which are linearly dependent on AB, BC, CD, DA. Since any point of Bl) is joined by lines of the family to A and C, it is joined by lines of 107 J THE LINEAll CONGRUENCE 315 tho family to every point of AC. Hence this case gives the family of all lines meeting both AC and BD. li\ caiso ft, 0, d determine a bundle of lines, a, being independent of them, does not pass through the center of the bundle. Hence the family of dependent lines includes all lines of the plane of a and the center of the bundle as well as the bundle itself. Lastly, if &, c, d are coplanar, we have, by duality, the same case as if I, e, d were concurrent. We have thus proved THEOREM 14. A linear congruence is cither (1) a set of lines linearly dependent on four linearly independent skew lines, such that no otic of them meets tlic rcgulus containing the other three in a proper point; or (2) it is the set of all lines meeting two skew lines; or (3) it is the set of all rulers and tangent lines of a given regulus which meet a Ji<ml directrix of the regulus ; or (4) it consists of a bundle of lines and a plane of lines, the center of the bundle being on the plane. DEFINITION. A congruence of the first kind is called elliptic; of the second kind, hyperbolic ; of the third kind, parabolic ; of the fourth kind, degenerate. A line which has points in common with all lines of a congruence is called a directrix of the congruence. COHOLLAKY. A parabolic congruence consists of all lines on corre- sponding points and planes in a projectivity between the points and planes on a line. The directrix is a line of the congruence. To study the general nondegenerate case, let us denote four linearly independent and mutually skew lines on which the other lines of the congruence depend by a, b, c, d, and let TT I and 7r 2 be two planes in- tersecting in a. Let the points of intersection with ^ and ?r 2 of b, c, and d be 7^, C v and D l and J? 2 , O v and > z respectively. By letting the complete quadrilateral a, B&, G^ D^ correspond to the complete quadrilateral a, J? 2 (7 2 , 2 Z> 2 , Z> 2 # 2 , there is established a protective collineation II between the planes ir^ and TT Z in which the lines &, a, d join homologous points (fig. 113). Among the lines dependent on a, b, c, d are the lines of the reguli ale, acd, adb, and all reguli containing a and two Knes from any of these three reguli. But all such reguli meet ^ and TT Z in lines (e.g. B^D V ^ 2 # 2 ) because they have a in common with ^ and 7T 2 , Furthermore, the lines of the fundamental reguli join points 316 FAMILIES OF LIKES which correspond in II (Theorem 5 of this chapter and Theorem 18, Chap. IV), Hence the reguli which contain a and linos nhown by means of such reguli to be dependent on a, i, c, d arc those gen- erated by the projectivities determined by II between lines of TT JL and 7r 2 . d fi* \ rio. 113 Now consider reguli containing triples of the lines already shown to be in the congruence, but not containing a. Three such linos, /, m, n, join three noncollinear points L v M lt ^ of ir l to the points L 2 , M ZJ jV 2 of 7r 3 which correspond to them in the collineation II. The regulus containing Z, m> and n meets TT^ and TT S in two conies which are protective in such a way that L v M v N^ correspond to &, Jf 2 , JV 9 . The projectivity between the conies determines a projectivity between the planes, and as this projectivity has the same effect an II on the quadrilateral composed of the sides of the triangle L^M^ and the line a, it is identical with II. Hence the lines of the regulus I, m, n join points of 7r t and 7r 2 which are homologous under II and are therefore among the lines already constructed. Among the lines linearly dependent on the family thus far con- structed are also such as appear in flat pencils containing two inter- secting lines of the family. If one of the two lines is a, the other must meet a in a double point of the projectivity determined on a by II. If neither of the two lines is a, they must meet 7r a and 7T 3 , the first in points JFJ, JEJ and the second in points Q v <? 2 , and these four 107J THE LINEAK CONGRUENCE 317 points are clearly distinct from one another. But as the given lines of the congruence, JJJJ and Q^, intersect, so must also the lines P^ and Wa ' rrr \ an( l ^a intersect, and the projectivity determined between l[Qi and Jj() a by II is a perspectivity. Hence the common point of ./^ and JJ(J a IH a point of # and is transformed into itself by II. Hence, if lines of the family intersect, II has at least one double point on a, which means, by 105,* that the line a meets the regulus led and the congruence has one or two directrices. Thus two lines of a nondcgenerate congruence intersect only in the parabolic and hyper- bolic cases ; and from our previous study of these cases we know that the lines of a congruence through a point of intersection of two lines form a flat pencil. We have thus shown that all the lines linearly dependent on a, I, c, d, with the exception of a flat pencil at each double point of the projectivity on a, are obtained by joining the points of TT X and 7r 2 which are homologous under II. From this it is evident that any four linearly independent lines of the congruence could have been taken as the fundamental lines instead of a, 6, c, d. These two results are summarized as follows : THKOKKM 15. All the lines of a linear congruence are linearly dependent on any linearly independent four of its lines. No lines not in the congruence are linearly dependent on four such lines. THEOREM 16. If two planes meet in a line of a linear congruence and neither contains a directrix, the other lines of the congruence meet the planes in homologous points of a projectivity. Conversely, if two planes are protective in such a way that their line of intersection cor- responds to itself, the lines joining homologous points are in the same linear congruence. * If there are two double points, J0, JP, on a, the conic BiCiDiEF must be trans- formed by H into -the conic BaOtDtEFi and the lines joining corresponding points of these conies must form a regulus contained in the congruence. As JS? and 3? are on lines of the regulus 6c<Z, there are two directrices jp, q of this regulus which meet *$ and JP respectively. The lines p and g meet all four of the lines a, &, c, d. Hence they meet all lines linearly dependent on a, 6, c, d. In the parabolic case the regulus bed must be met by a in the single invariant point IT of the parabolic projectivity on a, because the conic tangent to a at IT and passing through BtCiDi must be transformed by II into the conic tangent to a at JET and passing through JfeCiD* ; and the'lines joining homologous points of these conies must form a regulus contained in the congruence. As JET, a point of a, is on a line of the regulus &ctf, there is one and only one directrix p of this regulus which meets all four of a, 6, c, d and hence meets all lines of the congruence. 318 FAMILIES OF LINES [CHAIN XI The dual of Theorem 16 may be stated in the following form: THEOREM 17. From two points* on the stmw line, of a, linear congru- ence the latter is projected Ity two projecMoe "biuutlM of pitmen, (^in- versely, two bundles of planes projectile in such a wj/ Unit the. line, joining their centers is self -corresponding, generate a Iw.ar MnyruMiec. DEFINITION. A regulus all of whose rulers are in, a congruence is called a regulus of-the congruence and is said to le in or to le con- tained in the congruence. COROLLARY. If three lines of a regulus are in a congruence, the regulus is in the congruence. In the hyperbolic (or parabolic) case the regulus led (in the notation already used) is met by a in two points (or one point), its points of intersection with the directrices (or directrix). In the elliptic case the regulus bed cannot be met by a in proper points, because if it were, the projectivity II, between TT X and 7r 2 , would have these points as double points. Hence no line of the congruence meets a regulus of the congruence without being itself a generator. Hence through each point of space, without exception, there is one and only one line of the congruence. The involution of conjugate, points of the regulus led on the line a is transformed into itself by II, and 'the same must be true of any other regulus of the congruence, if it does not con- tain a. Since there is but one involution transformed into itself by a noninvolutoric projectivity on a line (Theorem 20, Chap. VIII), we have that the same involution of conjugate points is determined on any line of the congruence by all reguli of the congruence whicli do not contain the given line. This is entirely analogous to the hyper- bolic case, and can be used to gain a representation in terms of proper elements of the improper directrices of an elliptic congruence* The three kinds of congruences may be characterized as follows : THEOREM 18. In a parabolic linear congruence each line is tangent at a fixed one of its points to all reguli of the congruence of which it u not a ruler. On each line of a hyperbolic or elliptic congruence all reguli of the congruence not containmg the given line determine the same involution of conjugate points. Through each point of space there is one and only one line of an elliptic congruence, for hyperbolic and parabolic congruences this statement i$ true except for points on a directrix. i<>7, iosj TJIE LINEAR COMPLEX 319 EXERCISES 1. All lines of a congruence can bo constructed from four lines by means of reguli all of which have two given lines in counnon. 2. (liven two involutions (both having or both not having double points) ou two skew line,H. Through each point of apace there are two and only two lines which are axes of pern] activity projecting one involution into the other, i.e., Much that two planes through conjugate pairs of the first involution pass through a conjugate pair of the second involution. These lines constitute tWO COngrUetKHiH. 3. All lines of a congruence meeting a line not in the congruence form a regular, 4. A linear congruence i self-polar with regard to any regulus of the congruence. 5. A degenerate linear congruence consists of all linos meeting two inter- secting linos. 108* The linear complex. THTCOKEM 19. A linear complex con- mix of ail linw linearly dependent on the edges of a simple skew Proof, By definition ( 106) the complex consists of all lines linearly dependent on five independent lines. Let a be one of these which does not meet the other four, V, c f , $, e f . The complex consists of all linos dependent on # and the congruence Vctdld. If this con- gruence is degenerate, it consists of all lines dependent on three sides o a triangle ode and a line I not in the plane of the triangle (Theorems 14, 15). As 6 may be any line of a bundle, it may be chosen so as to meet a c may be chosen so as to meet I, and e may be so chosen as to meet a. Thus in this case the complex depends on five lines a, J, c, d, e not all coplanar, forming the edges of a simple pentagon. If the congruence is not degenerate, the four lines 6", c rr , d", e" upon which it depends may (Theorem 15) be chosen so that no two of them intersect, but so that two and only two of them, W and e n , meet a. Thus the complex consists of all lines linearly dependent on the two flat pencils aW and ae" and the two lines c" and d". Let & and e be the lines of these pencils (necessarily distinct from each other and from a) which meet c n and d" respectively. The complex then consists of all lines dependent on the flat pencils e&&, "be", ae, ed". * The edges of a simple skew pentagon are five lines in a given order, not all coplanar, each line intersecting its predecessor and the last meeting the first. 20 FAMILIES OF LINKS [(iiur. xi .Finally, let c and d be two intersecting linos distinct from // ami f, which are in the pencils bo n and #F'. The complex consists of all lines linearly dependent on the flat pencils (tf> 9 be-, cd, <l< l t w. Not all tlio vertices of the pentagon dbcdc can be ooplanar, because then all the lines would be in the same degenerate congruence. THEOREM 20. DEFINITION. There are two dawn of wm>j*lc t m #m'h that all contylettes of cither clans are projecti'veli/ MjKiwtleuf. A c<nn, plc& of one class consists of a line and all linw of xptw, which, meet it. These are called special complexes. A complex of the, ofhe.r cJtttttt is catted general. No four vertices of a pentagon which determines it are coplanar. Proof. Given any complex, by the last theorem there is at least one skew pentagon abode which determines it. If there is a line / meeting the five edges of this pentagon, this line must meet all lines of the complex, because any line meeting three linearly independent lines of a regains (degenerate or not) meets all lines of it. Moreover, if the line I meets a and I as well as c and rf, it must either join their two points of intersection or be the line of intersection of their common planes. If I meets e also, it follows in either case thai four of the vertices of the pentagon are coplanar, two of them being on c. (That all five cannot be coplanar was explained at the end of the last proof.) Conversely, if four of the five vertices of the skew pentagon are coplanar, two and only two of its edges are not in this plane, and the line of intersection of the plane of the two edges with the plane of the other three meets all five edges. Hence, if and only if four of the five vertices are wplwmr, there. r,r- ists a line meeting the Jive linen. Since any two skew pentagons are pro- jectively equivalent, if no four vertices are coplanar (Theorem 1U, Chap. Ill), any two complexes determined by such pentagons are protectively equivalent. Two simple pentagons are also equivalent if four vertices, but not five, of each are coplanar, because any simple planar four-point can be transformed by a collineation of space into any other, and then there exists a collineation holding the plane of the second four-point pointwise invariant and transforming any point not on the plane into any other point not on the plane. There- fore all complexes determined by pentagons of this kind are pro jec- tively equivalent. But these are the only two kinds of skew pentagons. Hence there are two and only two kinds of complexes. f THE LINE All COMPLEX 321 hi ease four vertices of the pentagon are coplanar, we have seen that there is a line / meeting all its edges. Since this line was detiH'mmed as the intersection of the plane of two adjacent edges with the plane of the other three, it contains at least two vertices. It ('.winot contain three vertices because then all five would be <*.o]>luiwr. AH one of the two planes meeting on I contains three independent lines, all lines of that plane are lines of the complex. Tim line / ilnel is therefore in the complex as well as the two lines of the other plane. Hence all lines of both planes are in the complex. II once all lines meeting / are in the complex. But as any regulus three of whose lines meet / has all its lines meeting Z, the complex IioM the requirements stated in the theorem for a special complex. Fi<*. 114 A more definite idea of the general complex may be formed as follows. Let piPiPdW* (% 11 ' i ) ke a simple pentagon upon whose edges all lines of the complex are linearly dependent. Let q be the line of the flat pencil jp 8 jp 4 which meets p v and let ft be the point of intersection of q and p r Denote the vertices of the pentagon by P 12 , & ^H> / ?&i> ^ ie subscripts indicating the edges which meet in a given vertex. The four independent lines p^^y determine a congruence of lines all of which are in the complex and whose directrices are # = jftj^ 8 and a f a* 7f 2 / 8 V Iu like manner, gp^p 6 p^ determine a congruence whose directrices are l RP^ and V^=P M P 6l . The complex consists of all lines linearly dependent on the lines of these two congruences. The 322 FAMILIES OF LINES [('HAP. XI directrices of the two congruences intersect at R and /,5 t respectively and determine two planes, al = p and aW^vr, which meet OH <y. Through any point P of space not on /> or TT then 1 , are two linos 1 9 m, the first meeting a and a r , and the second meeting h and // (fig. 115). All lines in the ilat pencil Ini are in the complex by deii- nition. This fiat pencil meets p and TT in two perspective ranges of FIG. 115 points and thus determines a projectivity between the flat pencil a<b and the flat pencil a'V 9 in which a and a f , & and V correspond and q corresponds to itself. The projectivity thus determined between the pencils ab and aV is the same for all points P> because a< 9 ft, q always correspond to a r , V, q f . Hence the complex contain,* all lines in the flat pencils of lines which meet homologous lines in the proJMtiwity determined ly alq -ft a'Vq. Denote this set of lines ly S. We have seen that it has the properly that all its lines through a point not on p or TT are coplanar. If a point P is on p but not on q, the line Pit has a corresponding linojp/ in the pencil a'V and hence S contains all lines joining /* to points of p f . Similarly, for points on TT bxit not on q. By duality every plane not on q contains a flat pencil of lines of S. Each of the flat pencils not on q has one line meeting q. Hence each plane of space not on q contains one and only one line of S meeting j. Applying this to the planes through P H not contain- ing 2, we have that any line through P^ and not on p is not in the i 108 J THE LINEAR COMPLEX 323 sot S. Let I be any such line. All lines of S in each plane through / form a flat pencil P, and the centers of all these pencils lie on a line /', because all lines through two points of I form two flat pencils each of which contains a line from each pencil P. Hence the lines of S meeting I form a congruence whose other directrix V evidently lies on p. The point of intersection of V with q is the center of a flat pencil of lines of S all meeting L Hence all lines of the plane lq form a flat pencil. Since I is any line on P 34 and not on TT, this establishes that uaeh plane and, by duality, each point on q, as well as not on q, con- tains a Hat pencil of lines of S. We can now prove that S contains no lines not in the complex. To do 80 we have to show that all lines linearly dependent on lines of S are in S. If two lines of S intersect, the flat pencil they deter- mine is by definition in S. If three lines m v m 29 m B of S are skew to one another, not more than two of the directrices of the regulus con- taining them are in S. For if three directrices were in S, all the tan- gent line** at points of these three lines would be in S, and hence any piano would contain three nonconcurrent lines of S. Let I be a directrix of the regulus m^m^m^ which is not in S. By the argu- ment made in the last paragraph all lines of S meeting I form a con- gruence. But this congruence contains all lines of the regulus mjn^n^ and hence all lines of this regulus are in S. Hence the set of lines ,S is identical with the complex. TIIKOREM 21 (SYLVESTER'S THEOREM*). If two protective flat pencils with different centers and planes have a line q in common which is Mlfaorretipontliny, all lines meeting homologous pairs of lines in these two pencils are in the same linear complex. This complex consists of these lines together with a parabolic congruence whose directrix is q. Proof. This has all been proved in the paragraphs above, with the exception of the statement that q and the lines meeting q form a linear congruence. Take three skew lines of the complex meeting q ; they determine with q a congruence C all of whose lines are in the complex. There cannot be any other lines of the complex meeting q, because there would be dependent on such lines and on the congru- ence C all lines meeting q, and hence all lines meeting q would be in the given complex, contrary to what has been proved above. * Cf. Coinptes Kendus, Vol. LH (1861), p. 741. 324 FAMILIES OF LINKS [CUAI>.XI Another theorem proved in the discussion above is: THEOREM 22. DEFINITION OF NULL MYWTKM. All the lhw& of a linear complex which pans through a point, /* lie, in <i plans. TT, and all the lines which lie in a plane TT jprwwf tlirowgh a 'point /*. In MM, of a special complex, exception must be made of the points and pht.nw on the directrix. The point P is called the null point of the plane TT and TT is called the null plane of P with regard to the wm-pliw. The cor- res^^ondence between the points and planes of space thus established is called a null system or null polarity. Another direct consequence, remembering that there are only two kinds of complexes, is the following : THEOREM 23. Any five linearly independent lines are in one and only one complex. If the edges of a simple pentagon are in a given complex, the pentagon is skew and its edges linearly independent. If the complex is general, no four vertices of a simple pentagon of its lines are eoplanar. THEOREM 24. Any set of lines, K, in space such that the lines of the set on each point of space constitute a flat pencil is a linear complex. Proof, (a] If two lines of the set K intersect, the set contains (ill lines linearly dependent on them, by definition. (6) Consider any line a not in the given set K. Two points A, B on a have flat pencils of lines of K on different planes ; for if the planes coincided, every line of the plane would, by (a), be a line of K. Hence the lines of K through A and B all meet a line a 1 skew to a. Prom this it follows that all the lines of the congruence whoso directrices are a, a' are ii^ K. Similarly, if & is any other line not in K but meet- ing a, all lines of K which meet & also meet another line ?/. More- over, since any line meeting a, &,and V is in K and hence also meets a 1 , the four lines a, a 1 , &, V lie on a degenerate regains consisting of the flat pencils aa r and W (Theorem 13). Let q (fig. 115) bo the common line of the pencils al and a!V. Through any point o sjwce not on one of the planes ab and a'U there are three eoplanar lines of K which meet q and the pairs aa r and W. Hence K consists of lines mooting homologous lines in the projectivity and therefore is a complex by Theorem 21. THE LINEAR COMPLEX 325 OOROLLAKY. Any (1,1) correspondence between the points and the planes of space such that each point lies on its corresponding plane in a null system. THEOREM 25. Two linear complexes have in common a linear congruence. Proof, At any point of space the two flat pencils belonging to the two complexes have a line in common. Obviously, then, there are three linearly independent lines l v L 2 , Z 3 common to the complexes. All lines in the regulus / i y a are, by definition, in each complex. But as there are points or planes of space not on the regulus, there is a line 1 4 common to the two complexes and not belonging to this regulus. All lines linearly dependent on l v Z 2 , Z 3 , 1 4 are, by definition, common to the complexes and form a congruence. No further line could be common or, by Theorem 23, the two complexes would be identical COROLLARY 1. The lines of a complex meeting a line I not in the complex form a hyperbolic congruence. Proof. The line is the directrix of a special complex which, by the theorem, has a congruence in common with the given complex. The common congruence cannot be parabolic because the lines of the first complex in a plane on I form a flat pencil whose center is not on I, since I is not in the complex. COROLLARY 2. The lines of a complex meeting a line I of the com- plex form a parabolic congruence. Proof. The centers of all pencils of lines in this congruence must be on I because I is itself a line of each pencil. DEFINITION. A line I is a polar to a line V with regard to a complex or null system, if all lines of the complex meeting I also meet V. COROLLARY 3. If I is polar to I', V is polar to I A line is polar to itself, if and only if it is a line of the complex. THEOREM 26. A null system is a protective correspondence "between the points and planes of space. Proof. The points on a line I correspond to the planes on a line V by Corollaries 1 and 2 of the last theorem. If I and V are distinct, the correspondence between the points of I and planes of V is a per- spectivity. If l*=V, the correspondence is projective by the corollary of Theorem 14. 320 FAMILIES OF LINES l<'n.u>. XI EXERCISES 1. If a point P is on a plane p, the null piano TT of /'is on the null point It of p. 2. Two pairs of lines polar with regard to the same, null system are uhvuys in the same regulus (degenerate, if a line of one, pair meets a lino of the other pair). 3. If a lino / meets a line m, the polar of I moots the polar of nt. 4. Pairs of lines of the, regains in Ex. "2 which are, polar with regard to the complex are met by any directrix of the regains in pairs of points of an involution. Tims the complex determines an involution among the linos of the regnlus. 5. Conversely (Theorem of Chasie,s), the lines meeting conjugate pairs of lines in an involution on a regulus are in the same complex. Show that Theorem 21 is a special case of this. 6. Find the lines common to a linear complex and a regulus not in the complex. 7. Three skew lines , I, m determine one and only one complex contain- ing k and having I and m as polars of each other. 8. If the number of points on a line is + 1, how many reguli, how many congraences, how many complexes are there in space? How many lines are there in each kind of regulus, congruence, complex ? 9. Given any general complex and any tetrahedron whose faces are not null planes to its vertices. The null planes of the vertices constitute a second tetrahedron whose vertices lie on the planes of the first tetrahedron. The two tetrahedra are mutually inscribed and circumscribed each to the, other* (cf. Ex. G, p. 105). 10. A null system is fully determined by associating with the tlirco vertices of a triangle three planes through these vertices and having their ono common point in the plane of the triangle but not on one of its sides. 11. A tetrahedron is self-polar with regard to a null system if two opposite edges are polar. 12. Every line of the complex determined by a pair of Mobius tolrahodra meets their faces and projects their vertices in projectivo throws of points and planes. 13. If a tetrahedron T is inscribed and circumscribed to 7\ and also to 7^, the lines joining corresponding vertices of !Z\ and T, 2 and the linos of intersec- tion of their corresponding planes are all in the same complex. 14. A null system is determined by the condition that two pairs of lines of a regulus shall be polar. 15. A linear complex is self-polar with regard to a regnlus all of whoso lines are in the complex. 16. The lines from which two projective pencils of points on skew linos are projected by involutions of planes are all in the same complex* Dunllxo. * This configuration was discovered by Mobius, Journal fur Mathomatik, Vol. HI (1828), p. 273. Two tetrahedra in this relation are known as Mobius tctrahwlra, iw] LINE COORDINATES 327 109. The Pliicker line coordinates. Two points whose coordinates are determine a line L The coordinates of the two points determine six numbers which are known as the Pluckw coordinates of the line. Since the coordinates of the two points are homogeneous, the ratios only of the numbers p lf are determined. Any other two points of the line deter- mine the same set of line coordinates, since the ratios of the jp y 's are evidently unchanged if (& v x, # 3 , # 4 ) is replaced by (x l + \y v o? 2 - The six numbers satisfy the equation* 0. Thin is evident on expanding in terms of two-rowed minors the identity x l 2/4 = o. (Jbnversely, if any six numbers, jp , are given which satisfy Equa- tion (1), then two points P^fa, %%> 8 , 0), Q~(y v 0, y 8 , y 4 ) can be determined such that the numbers p tf are the coordinates of the line PQ. To do this it is simply necessary to solve the equations which are easily seen to be consistent if and only if Hence we have THEOREM 27. Every line of space determines and is determined ly the ratios of six niwibers jp u , jp 18 , jp 14> p$v p&, p 2S subject to the * Notice that in Equation (1) the number of inversions in the four subscripts of any term is always even. 328 condition FAMILIES OF LINES [<',HAI>, XI ! 0, Mtch thttt. if (,','p ,-r5 a , .", , 4 ) <r/f it; 8 ^'4 ^ y/4 i'w ^=~" J'u- '*'4 ?/,! OOROLLAEY. Four independent coordinates determine, a line. In precisely similar manner two planes (u, v n^ w s , v^) and (v\, 7? a , v rt , v 4 determine six numbers such that 2 12 == M, The quantities 2 V . satisfy a theorem dual to the one just proved for the jp v 's. THEOREM 28. I7^e jp and % coordinates of a line are connected 1>y the equations jp 12 : _p ls : jp u : p 34 : ^ 42 : ^ 23 = J M : <7, 2 : JJT W : y la : </ u : y w . Proof. Let the j? coordinates he determined l)y the two points (x v XM x 3 , x^j (y v y z , y^ y 4 ), and the q coordinates by the two planes (v, v u# u a , u 4 ), (v v v> 2> -z; 8 , i? 4 ). These coordinates satisfy the four equations u^ + Ufa + Ufa + Ufa = 0, V& + vfa + vfa + v& = 0, ^2/1 + ^2/2 + ^ 8 + ^4 = 0, ^1^1 + ^8 + ^#8 + '^4^4 = ' Multiplying the first equation by # x and the second by W L and adding, we obtain In like manner, from the third and fourth equations wo obtain Combining the last two equations similarly, we obtain or, * = *. <?14 Pi* By similar combinations of the first four equations we find i* n u : Pu : : Si* : io, noj LINE COORDINATES 329 EXERCISE Given tlie tetrahedron of reference, the point (1, 1, 1, 1), and a line I, determine six sets of four points each, whose cross ratios are the coordinates of/. 110. Linear families of lines. THEOREM 29. The necessary and tni,f/i('.ieti,t condition that two lines p and p f intersect, and hence are coplanar, is where p (j are the coordinates of p and p!j of p 1 . Proof. If the first line contains two points x and y, and the second two points x 1 and y r , the lines will intersect if and only if these four points are coplanar ; that is to say, if and only if = ffi ft ?/ ./ ,/ ,/ '''i '^a '*'8 ';/' <?/' <?/' yi 2^2 ya uP* + J'sJ'u TIIEOHKM 30. Aflat pencil of lines consists of the lines whose coordi- nates are Xp,. ; +/*p^> if p und p f are two lines of the pencil. Proof, The lines p and p f intersect in a point A and are perspec- tive with a range of points <7+XZ>. Hence their coordinates may be written which may be expanded in the form THEOREM 31. The lines whose coordinates satisfy one linear eqitation form a linear complex. Those whose coordinates satisfy two independ- ent linear equations form a linear congruence, and those satisfying three independent linear equations form a regulus. Four independent linear equations are satisfied ly two (distinct or coincident] Iines 9 which may 'be improper. 330 FAMILIES OF LINES [CHAP. XI Proof. If (\, Z^, J 8 , 6 4 ) is any point of space, the points (M V w a , ,f,,, ,f 4 ) which lie on lines through 1 19 J a , 6 8 , 6 4 satisfying (1) must satisfy KX.X^^X,^ ( '4 '"'2 = 0, or (2) ^ A) t/; a 4 which is the equation of a plane. Hence the family of lines repre- sented by (1) has a flat pencil of lines at every point of space, and so, by Theorem 24, is a linear complex. Since two complexes have a congruence of common lines, two linear equations determine a congruence. Since a congruence and a complex have a regulus in common, three linear equations determine a regulus. If the four equations are independent, one of the four-rowed determinants of their cooili- cients is different from zero, and the equations have solutions of the If one of these solutions is to represent the coordinates of' a line, it must satisfy the condition which gives a quadratic equation to determine X//A. Hence, by Propo- sition K 2 , there are two (proper, improper, or coincident) lines whose coordinates satisfy four linear equations. COKOLLARY 1. TJie lines of a regulus are of the form where p r , p fr , p nr are lines of the reyulus. In like manner, (lie Un&x of a congruence are of the form * Cf . Bocher, Introduction to Higher Algebra, Chap. IV. >, m] LINE COORDINATES 331 of tt complex of the form All of these formulas must le taken in connection with CouoLLAiiY 2. As a transformation from points to planes the null determined "by the complex whose equation is = / -y xv m I C\ The first of these corollaries simply states the form of the solu- tions of systems of homogeneous linear equations in six variables. The second corollary is obtained by inspection of Equation (2) the coefficients of which are the coordinates of the null plane of the point (l v & 2 , 5 8 , & 4 ). Corollary 1 shows that the geometric definition of linear dependence of lim*H givon in thin chapter corresponds to the conventional analytic concep- tion of linear dependence. 111. Interpretation of line coordinates as point coordinates in S 5 . It may be shown without difficulty that the method of introducing homogeneous coordinates in Chap. VII is extensible to space of any number of dimensions (cf. Chap. I, 12). 'Therefore the set of all sets of six numbers can be regarded as homogeneous point coordinates in a space of five dimensions, S 6 . Since the coordinates of a line in S s satisfy the quadratic condition they may be regarded as forming the points of a quadratic locus or spread,* L 4 a , in S 6 . The lines of a linear complex correspond to the points of intersection with this spread of an S 4 that is determined by one linear equation. The lines of a congruence correspond, therefore, to the intersection with L 4 2 of an S 3 , the lines of a regulus to the * This is a generalization of a conic section. 332 FAMILIES OF LINES [<hui>. xi intersection with Lj 2 of au S 2 , and any pair of linos to the intersee- tion with L* of an S r Any point (^ p(. l9 p{, |4, }>&,}>&) of S c has as its polar*' S 4 , with regard to L 4 2 , (2) l*Ll\i + pJaPu + 2>*2>u + P/a^M + 1>l* Vi* + u ?' = (] > .which is the equation of a linear complex in the original S. r any point in S 5 cttn be thought of tin re.preMnting the, winylcs, of I represented Inj the points of S 5 in whitfi its poltir S 4 meets L.j*. Since a line is represented by a point on L 4 a , a special complex is represented by a point on L^ 2 , and all the lines of the special complex by the points in which a tangent S 4 meets L*. The points of a line, # + X&, in S 5 represent a set of complexes whose equations are (3) ( and all these complexes have in common the congruence common to the complexes a and &. Their congruence, of course, consists of the lines of the original S 3 represented by the points in which L* is met by the polar S 3 of the line a + X&. A system of complexes, a + \l, is called a pencil of ctMi'pliWx, and their common congruence is called its Iww or laml MMfrwnw. It evidently has the property that the null planes of any point with regard to the complexes of the pencil form an axial pencil whose axis is a line of the basal congruence. Dually, the null points of any plane with regard to the complexes of the pencil form a rnngo of points on a line of the basal congruence. The cross ratio of four complexes of a pencil may be defined as the cross ratio of their representative points in S 6 . From the. form of Equation (3) this is evidently the cross ratio of the four null planes of any point with regard to the four complexes. A pencil of complexes evidently contains the special complexes whose directrices are the directrices of the basal congruence. Ilenco * Equation (2) may be taken as the definition of a polar S 4 of a point with regard to L|. Two points are conjugate with regard to L* if the polar S 4 of one contains the other. The polar S 4 's of the points of an S t - (i =s 1, 2, &, 4) all have an S 4 _, in common which is called the polar S 4 ^ t - of the S;. These and other obvioiw generalizations of the polar theory of a conic or a regains we take for granted without further proof. 1J J LIKE COOEDINATES 333 there are two improper, two proper, one, or a flat pencil of lines which are the directrices of special complexes of the pencil. These cases arise as the representative line a + \l meets L 4 2 in two improper points, two proper points, or one point, or lies wholly on L 4 2 . Two points in which a representative line meets L 2 are the double points of an involution the pairs of which are conjugate with regard to L 2 . Two complexes p, p r whose representative points are conjugate with regard to L.* are said to be conjugate or in involution. They evidently satisfy Equation (2) and have the property that the null points of any plane with regard to them are harmonically conjugate with regard to the directrices of their common congruence. Any complex a is in involution with all the special complexes whose directrices are lines of a. Let ff t be an arbitrary complex and & 2 any complex conjugate to (in involution with) it. Then any representative point in the polar S 8 with regard to L* of the representative line a^a 2 represents a complex conjugate to a l and a y Let cr, B be any such complex. The represent- ative points of a v a a , a s form a self-conjugate triangle of L 4 2 . Any point of the representative plane polar to the plane a^a^ with regard to L 4 2 is conjugate to a^a^ Let such a point be 4 . In like manner, a. $ and & can be determined, forming a self-polar 6-point of LJ*, the generalization of a self-polar triangle of a conic section. The Hix points are the representatives of six complexes, each pair of which in in involution. It can be proved that by a proper choice of the six points of refer- ence in the representative S e , the equation of L 4 2 may be taken as any quadratic relation among six variables. Hence the lines of a three- space may be represented analytically by six homogeneous coordinates subject to any quadratic relation. In particular they may be repre- sented by (#i, a? 9 ,- , # 6 ), where = 0.* In this case, the six-point of reference being self-polar with regard to L*, its vertices represent complexes which are two by two in involution. * Those are known as Klein's coordinates. Most of the ideas in the present sec- tion are to be found in F. Klein, Zur Theorie der Liniencomplexe des ersten zwoiten Grades, Mathematische Annalen, Vol. II (1870), p. 198. 334 FAMILIES OF LINKS Loiur. xi EXERCISES 1. If a pencil of complcxe-H contains two special complexes, the. basal con- gruence of the pencil is hyperbolic or elliptic, according as the special com- plexes are proper or improper. 2. If a pencil of linear complexes contains only a single special complex, the basal congruence is parabolic. 3. If all the complexes of a pencil of linear complexes are special, tlw basal congruence is degenerate, 4j. Define a pencil of complexes as the system of all complexes having a common congruence of lines and derive its properties synthetically. 5. The polars of a line with regard to the complexes of a pencil form a regulus. 6. The null points of two planes with regard to the complexes of a pencil generate two projective pencils of points. 7. If C = 0, C'= 0, C"' = are the equations of three linear complexes which do not have a congruence in common, the equation C + AC" + i**C" = is said to represent a "bundle of complexes. The lines common to the three fundamental complexes C, C", C" of the bundle form a regulus, the con- jugate regulus of which consists of all the directrices of the special com- plexes of the bundle. 8. Two linear complexes Sa, 7j fr/ = and Sfyjft/ = are in invohition if and only if we have + G * As 9. Using Klein's coordinates, any two complexes are given by and SMi 0. These two are in involution if 2a^ = 0. 10. The six fundamental complexes of a system of Klein's coordinates intersect in pairs in fifteen linear congruences all of whose directrices are dis- tinct. The directrices of one of these congruences are lines of tho remaining four fundamental complexes, and meet, therefore, the twelve directrices of the six congruences determined by these four complexes. INDEX The numbers refer to pages Abolian group, 67 Absoksa, 170 Abstract science-, 2 Addition, of points, 142, 231 ; theorems on, 142-144 ; other definitions of, 1C7, Em 3, 4 Adjacent sides or vertices of simple ft-lino, 87 Algebraic curve, 259 Algebraic problem, 288 Algebraic surface, 259 Alignment, assumptions of, 10 ; consist- ency of assumptions of, 17; theorems of, for the, plane, 17-20 ; theorems of, for #-space, 20-24 ; theorems of, for 4-space, 25, Ex. 4; theorems of, for n-spaco, 29-33 Amodeo, E,, 120, 294 Anharmonic ratio, 159 Apollonius, 286 Associative law, for correspondences, 06 ; for addition of points, 143 ; for null tit >li cation of points, 146 Assumption, H , 45; H , rOlo of, 81, 261; of projectivity, 95; of projec- tivity, alternative forms of, 105, 106, Exs. 10-12 ; 298 Assumptions, are necessary, 2; exam- ples of, for a mathematical science, 2; consistency of, 3; independence of, 6 ; categoricalnoss of, 6 j of align- ment, 16; of alignment, consistency of, 17; of extension, 18, 24; of clo- sure, 24 ; for an n-space, 33 Axial pencil, 55 Axial perspectivity, 57 Axis, of perspectivity, 36; of pencil, 55 ; of perspective collineation, 72 ; of homology, 104; of coSrdinates, 169, 191; of projectivity on conic, 218 Bawo, of piano of points or lines, 55 ; of pencil of complexes, 332 Bilinear equation, "binary, represents projectivity on a line, 156 ; ternary, represents correlation in a plane, 267 Binary form, 251, 252, 254 B&cuor, M., 156, 272, 289, 330 Braikenriclge, 119 Brian chon point, 111 Brianchon's theorem, 111 Bundle, of planes or lines, 27, 55; of conies, 297, Exs. 9-12; of quadrics, 311 ; of complexes, 334, Ex. 7 Burnside, W., 150 Bussey, W. H,, 202 Canonical forms, of collineations in plane, 274-276; of correlations in a plane, 281 ; of pencils of conies, 287- 293 Castelnuovo, G., 139, 140, 237, 297 Categorical set of assumptions, 6 Cayley, A., 52, 140 Center, of perspectivity, 36 ; of flat pen- cil, 55 ; of bundle, 55 ; of perspective collineation in plane, 72 ; of perspec- tive collineation in space, 75; of homolpgy, 104; of coordinates, 170; of projectivity on conic, 218 Central perspectivity, 57 Characteristic constant of parabolic projectivity, 207 Characteristic equation of matrix, 165 Characteristic throw and cross ratio, of one-dimensional projectivity, 205, 211, Exs. 2, 3, 4 ; 212, Exs. 5, 7 ; of involu- tion, 206; of parabolic projectivity,, 206 Chasles, 125 Class, notion of, 2 ; elements of, 2 ; re- lation of belonging to a, 2 ; subclass of a, 2 ; undefined, 15 ; notation for, 57 Clebsch, A., 289 Cogredient n-line, 84, Ex. 13 Cogredient triangle, 84, Exs. 7, 10 Collineation, defined, 71 ; perspective, in plane, 72 ; perspective, in space, 75 ; transforming a quadrangle into a quadrangle, 74 ; transforming a five- point into a five-point, 77 ; transform- ing a conic into a conic, 182 ; in plane, analytic form of, 189, 190, 268 ; be- tween two planes, analytic form of, 190 ; in space, analytic form of, 200 ; leaving conic invariant, 214, 220, 235, Ex. 2; is the product of two polar- ities, 265; which is the product of two reflections, 282, Ex. 5 ; double ele- ments of, in plane, 271 ; character- istic equation of, 272 ; invariant figure of, is self-dual, 272 335 336 INDEX Collineationw, types of, in plants 100, 278; associated with two conies oC a pencil, i;U, EXH. 2, 4, 0; 135, Kx. 2 ; 130, Ex. 2 ; group of, in piano, 208; represented by matriees, 208- 270; two, not in general commuta- tive, 208 ; canonical forms of, 274- 276 Commutative C'orrospondoncc, 00 Commutative group, 07, 70, Kx. 1; 228 Commutative law of multiplication, 148 Commutative pro jectivi ties, 70, 210, 228 Compass, constructions with, 240 Complete Wrline, in plane, 37; on point, 38 Complete w-plane, in space, 37 ; on point, 38 Complete n-point, in space, 30 ; in piano, 37 Complete quadrangle and quadrilat- eral, 44 Complex, linear, 312; determined by skew pentagon, 319 ; general and spe- cial, 320 ; determined by two projec- tive flat pencils, 323 ; determined by live independent lines, 324; deter- mined by correspondence between points and planes of space, 324 ; null system of, 324 ; generated by involu- tion on regains, 320, Ex. 6 ; equation of, 329, 331 Complexes, pencil of, 332; in involu- tion, 333 ; bundle of, 334, Ex. 7 Concrete representation or application of an abstract science, 2 Concurrent, 10 Cone, 118 ; of lines, 109 ; of planes, 109 ; section of, by plane, is conic, 109; as degenerate case of quadric, 308 Configuration, 38; symbol of, 38; of Desargues, 40, 51 ; quadrangle-quad- rilateral, 44; of Pappus, 98, 249; of Mobius, 320, Ex. 9 Congruence, linear, 312 ; elliptic, hyper- bolic, parabolic, degenerate, 316; de- termined by four independent lines, 317 ; determined by projective planes, 317; determined by two ^complexes, 325 ; equation of, 329, 330 Conic, 109, 118 , theorems on, 109-140 ; polar system of, 120-124; equation of, 185, 245; projectiyity on, 217; intersection of lino with, 240, 242, 240 ; through four points and tangent to line, 250, Ex. 8; through three points and tangent to two linos, 250, Ex. 9 ; through four points and moot- ing given line in two points harmonic with two given points, 250, Ex. 10; determined by conjugate points, 293, Ex. 2 ; 294, Exs. 3, 4 Conic section, 118 (Ionics, pencils and ranges of, 128-130, 287-293; projeetive, 212, ,*K)4 Conjugate groups, 209 Conjugate pair of involution, 102 Conjugate points (lines), with regard to conic, 122; on line (point), form invo- lution, 124; wifch regard to a pencil of monies, U)0, Bx. ;J ; 140, Kx, JU ; 293, Kx. 1 Conjugate projeotivitios, 208; condi- tions for, 208, 209 Conjugate subgroups, 211 Consistency, of a sot of assumptions, # ; of notion of elements at infinity, 9; of assumptions of alignment, 17 Construct, 45 Constructions, linear (first degree), 230 ; of second degree, 245, 249-250, Exs.; of third and fourth degrees, 294-290 Contact, point of, of lino of lino conic, 112; of second order between two conies, 134; of third order between two conies, 130 Conwell, G. M., 204 Coordinators, non homogeneous, of points online, 152; homogeneous, of points on lino, 103; nonhomogeneous, of points in plane, 109; iionhomogeno-. OIIK, of linos in plane, 170; homogene- ous, of point;* and linos in piano, 174; in a bundle, 179, Ex. 3; of quadran- gle-quadrilateral configuration, 181, Ex. 2; nonhomogoneous, in space, 190; homogeneous, in space, 394; Plttokor'H lino, 327 ; Klein's lino, MM Coplanar, 24 Copunctal, 10 Correlation, between two-dimensional forms, 202, 203; induced, 202; be- tween two-dimensional forum deter- mined by four pairs o homologous elements', 204; which interchanges vortices and sides of triangle is polar- ity, 204 ; between two pianos, analytic representation of, 200, 207; repre- sented by ternary bilinear form, 207 ; represented by matrices, 270; double pairs of a, 278-281 Correlations and duality, 20R Correspondence, as a logical term, 5j perspective, 12; (1, 1) of two liguros, 35; general theory of, 04-0(1; idim- tioal, 05; inverse of, 05; period of, (JO; periodic or cyclic, 00; involutorio or reflexive, 00 ; perspective between two pianos, 71 ; quadratic, 139, KXH. 22, 24; 293, Ex. 1 Correspondences, resultant or product of two, 05; associative law for, 00; commutative, 60 ; groups of, 07; leav- ing a figure invariant form a group, 08 INDEX 337 Corresponding elements, 35; doubly, 102 Co variant, 257; example of, 258 Oemona, L., 187, 138 OroKH ratio, 151); of harmonic set, 150, Hit ; definition of, 100 ; expression for, 100; iu UoiiH Aeneous coordinates, 105; theorems on, 107, 108, EXH. ; characteristic, of projectivity, 205; characteristic, of involution, 200 ; as an invariant of two quadratic binary forms, 251, Ex. 1 ; of four complexes, o>2 Crows ratios, the six, defined by four ele- ments, 101 Curve, of third order, 217, Exs. 7, 8, 0; algebraic, 250 Cyclic e,orrespondence, 00 Darboux, <}., 05 Degenerate conies, 120 Degenerate regulus, 311 "Decree of geometric problem, 230 Derivative, 255 Desargucs, configuration of, 40, 51 ; the- orem on perspective triangles, 41, 180; theorem on conies, 127, 128 Descartes, R., 285 Diagonal point (line), of complete quad- rangle (quadrilateral), 44; of com- plete u-point (n-line) in plane, 44 Diagonal triangle of quadrangle (quad- rilateral), 41 Dirkson, L. K., 00 Difference of two points, 148 Differential operators, 250 Dimensions, space of three, 20 ; space of n, 30 ; assumptions for space of ?i, 33 ; space of live, 331 "Directrices, of a rogulus, 200 ; of a con- gruence, 315; of a special complex, IJ24 Distributive law for multiplication with respect to addition, 147 Division of points, 140 Domain of rationality, 238 Double element (point, line, plane) of correspondence, 08 Double pairs of a correlation, 97 Double* points, of a projeetivity on a line satisfy a quadratic equation, 150 ; of projectivity on a lino, homogeneous coordinates of, 104; of projectivity always exist in extended space, 242 ; of projectivity on a line, construction of, 240 ; of involution determined by covariant, 258 ; and lines of collinea- tion in plane, 271, 205 Doublo ratio, 150 Doubly parabolic point, 274 Duality, in three-space, 28; in plane, 29 j at a point, 259 j in four-space, 29, Ex. ; a consequence of existence of correlations, 208 Edge of %-point or ?i-plane, 36, 37 Elation, in plane, 72 ; iu space, 75 Element, undefined, 1; of a figure, 1; fundamental, 1; ideal, 7; simple, of space, 34; invariant, or double, or lixed, 08 ; lineal, 107 Eleven-point, plane section of, 53, Ex. 15 Enriques, F., 50, 286 Equation, of line (point), 174; of conic, 185, 245; of plane (point), 193, 198; reducible, irreducible, 239 , quadratic, has roots in extended space, 242 Equivalent number systems, 150 Extended space, 242, 255 Extension, assumptions of, 18, 24 Face of ?i-point or n-plaue, 36, 37 format, P., 285 Field., 140 ; points on a line form a, 151 ; finite, modular, 201 ; extended, in which any polynomial is reducible, 200 Figure, 34 Fine, H. B., 255, 260, 261, 289 Finite spaces, 201 Five-point, plane section of, in space, 39 ; in space may be transformed into any other by projective collineation, 77 ; diagonal points, lines, and planes of, in space, 204, Exs. 16, 17, 18; simple, in* space determines linear congruence, 319 Five-points, perspective, in four-space, 54, Ex. 25 Fixed element of correspondence, 68 Flat pencil, 55 Forms, primitive geometric, of one, two, and three dimensions, 55 ; one-dimen- sional, of second degree, 109; linear binary, 251; quadratic binary, 252; of nth degree, 254 ; polar forms, 256 ; ternary bilinear, represents correla- tion in plane, 207 Four-space, 25, Ex. 4 Frame of reference, 174 Fundamental elements, 1 Fundamental points of a scale, 141, 231 Fundamental propositions, 1 Fundamental theorem of projectivity, 94-97, 213, 264 General point, 129 Geometry, object of, 1; starting point of, 1 ; distinction between projective and metric, 12 ; finite, 201 ; associated with a group, 259 Gergonne, J. D., 29, 123 Grade, geometric forms of first, second, third, 55 Group, 66 ; of correspondences, 67 ; gen- eral projective, on line, 68, 209; 338 INDEX examples of, 09, 70 ; commutative, 70 ; general project! ve, in plane, 268 HO, assumption, 45 ; r61e of, 81, 261 Harmonic conjugate, 80 Harmonic honiology, 223 Harmonic involutions, 224 Harmonic set, 80-82 ; exercises on, 83, 84 ; cross ratio of, 159 Harmonic transformations, 230 Harmonically related, 84 Hesse, 125 Hessenberg, G., 141 Hexagon, simple, inscribed in two inter- secting lines, 99 ; simple, inscribed in three concurrent lines, 250, Ex. 5; simple, inscribed in conic, 110, 111 Hexagram, of Pascal (hexagramma mys~ ticum), 138, Exs. 19-21; 304, Ex. 16 Hilbert, D., 3, 95, 148 Holgate, T. P., 119, 125, 139 Homogeneous coordinates in plane, 174 Homogeneous coordinates, in space, 11, 194 ; on line, 163 ; geometrical signifi- cance of, 165 Homogeneous forms, 254 Homologous elements, 35 Homplogy, in plane, 72; in space, 75; axis and center of, 104; harmonic, 223, 275 ; canonical lorm of, in plane, 274, 275 Hyperosculate, applied to two conies, 136 Ideal elements, 7 Ideal points, 8 Identical correspondence, 65 Identical matrix, 157, 269 Identity (correspondence), 65; element of group, 67 Improper elements, 239, 241, 242, 255 Improper transformation, 242 Improperly protective, 97 Independence, of assumptions, 6 ; neces- sary for distinction between assump- tion and theorem, 7 Index, of subgroup, 271 ; of group of col- lineations in general protective group in plane, 271 Induced correlation in planar field, 262 Infinity, points, lines, and planes at, 8 Inscribed and circumscribed triangles, 98, 250, Ex. 4 Inscribed figure, in a conic, 118 Invariant, of two linear binary forms, 252 ; of quadratic binary forms, 252- 254, Ex. 1; of binary form of nth degree, 257 Invariant element, 68 Invariant figure, under a correspond- ence, 67 ; of collineation is self -dual, 272 Invariant subgroup, 211 Invariant triangle of collineation, rela- tion between project! vities on, 274, 276, Ex. 5 Inverse, of a correspondence, 65; of element in group, 67; of projectiyity is a projectivity, 68; of projectivity, analytic expression for, 157 Inverse operations (subtraction, divi- sion), 148, 149 Involution, 102 ; theorems on, 102, 103, 124, 127-131, 133, 134, 136, 206, 209, 221-229, 242-243 ; analytic expression for, 157, 222, 254, Ex. 2 ; character- istic cross ratio of, 206 ; on conic, 222- 230 ; belonging to a projectivity, 220 ; double points of, in extended space, 242 ; condition for, 254, Ex. 2 ; dou- ble points of, determined by covari- ant, 258 ; complexes in, 333 Involutions, any projectivity is product of two, 223; harmonic, 224; pencil of, 225; two, have pair in common, 243; two, on distinct lines are per- spective, 243 Involutoric correspondence, 66 Irreducible equation, 239 Isomorphism, 6; between number sys- tems, 150 ; simple, 220 Jackson, D., 282 Join, 16 Kantor, S., 250 Klein, P., 95, 333, 334 Ladd, C., 138 Lage, Geometrie der, 14 Lennes, N. J., 24 Lindemann, F., 289 Line, at infinity, 8 ; as undefined class of points, 15 ; and plane on the same three-space intersect, 22 ; equation of, 174; and conic, intersection of, 240, 246 Line conic, 109 Line coordinates, in plane, 171 ; in space, 327, 333 Lineal element, 107 Linear binary forms, 251 ; invariant of, 251 Linear dependence, of points, 30; of lines, 311 Linear fractional transformation, 152 Linear net, 84 Linear operations, 236 Linear transformations, in plane, 187; in space, 199 Lines, two, in same plane intersect, 18 Liiroth, J., 95 INDEX 339 Maclaurin, C., 119 MacNeish, H. P., 46 Mathematical science, 2 Matrices, product of, 156, 268 ; determi- nant of product of two, 269 Matrix, as symbol for configuration, 38 ; definition, 150; used to denote pro- . jectivity, 15C; identical, 157, 269; characteristic equation of, 165, 272; conjugate, transposed, adjoint, 269; as operator, 270 MensBchnuw, 126 Metric geometry, 12 Midpoint of pair of points, 230, Ex. 6 Mbbius tetrahedra, 105, Ex. 6; 326, Ex. 9 Multiplication of points, 145, 231 ; the- orems on, 145-148 ; commutative law of, is equivalent to Assumption P, 148; other definitions of, 107, Exs. 3,4 n-lino, complete or simple, 37, 38; in- scribed in conic, 138, Ex. 12 nrplane, complete in space, 37 ; on point, 38 ; simple in space, 37 n-poinl, complete, in space, 36 ; complete, in a plane, 37; simple, in space, 37; simple, in a plane, 37; plane section of, in space, 53, Exs. 13, 16 ; 54, Ex. 18 ; m-space section of, in (n + ] )-space, 54, Ex. 19 , section by three-space of, in four-space, 54, Ex. 21; inscribed in conic, 119, Ex. 5; 250, Ex. 7 n-points, in different planes and per- spective from a point, 42, Ex. 2 ; in same plane and perspective from a line, 42, Ex. 4; two complete, in a plane, 53, Ex. 7 ; two perspective, in (w l)-spaco, theorem on, 54, Ex. 26 ; mutually inscribed and circum- scribed, 250, Ex. 6 Net of rationality, on line (linear net), 84; theorems on, 85; in plane, 86; theorems on, 87, 88, Exs. 92, 93; in space, 89 ; theorems on, 89-92, Exs. 92, 93 ; in plane (space) left invariant by perspective collineation, 93, Exs. 9, 10; in space is properly projective, 97; coordinates in, 162 Newson, II. B., 274 Nonhomogeneous coordinates, on a line, 162 ; in plane, 169 ; in space, 190 Null system, 324 Number system, 149 On, 7, 8, 15 Operation, one-valued, commutative, as- sociative, 141 ; geometric, 236 ; linear, 236 Operator, differential, 256 ; represented by matrix, 270 ; polar, 284 Opposite sides of complete quadrangle, 44 Opposite vertex and side of simple 7i-point, 37 Opposite vertices, of complete quadrilat- eral, 44 ; of simple ?i-point, 37 Oppositely placed quadrangles, 50 Order, 60 Ordinate, 170 Origin of coordinates, 169 Osculate, applied to two conies, 134 Padoa, A., 3 Papperitz, E., 309 Pappus, configuration of, 98, 99, 100, 126, 148 Parabolic congruence, 315 Parabolic point of collineation in plane, 274 Parabolic projectivities, any two, are conjugate, 209 Parabolic projectivity, 101; charac- teristic cross ratio of, 206; analytic expression for, 207 ; characteristic con- stants, 207 ; gives H(MA', AA"), 207 Parametric representation, of points (lines) of pencil, 182; of conic, 234; of regulus, congruence, complex, 330, 331 Pascal, B., 36, 99, 111-116, 123, 120, 127, 138, 139 Pencil, of points, planes, lines, 55; of conies, 129-136, 287-293; of points (lines), coordinates of, 181 ; paramet- ric representation of, 182 ; base points of, 182 ; of involutions, 225 ; of com- plexes, 332 Period of correspondence, 66 Perspective collineation, in plane, 71 ; in space, 75 ; in plane defined when center, axis, and one pair of homol- ogous points are given, 72 ; leaving R z (R*) invariant, 93, Exs. 9, 10 Perspective conic and pencil of lines (points), 215 Perspective correspondence, 12, 13 ; be- tween two planes, 71, 277, Ex. 20 Perspective figures, from a point or from a plane, 35 ; from a line, 36 ; if A, JB, C and A', B', C' on two coplanar lines are perspective, the points (AB', JR4/), (AC', CA^, and (BC f , CB') are collinear, 62, Ex. 3 Perspective geometric forms, 56 Perspective n-lines, theorems on, 84, Exs. 13, 14 ; five-points in four-space, 54, Ex, 25 Perspective (n + l)-points in w-space, 54, Exs. 20, 26 Perspective tctrahcdra, 43 Perspective triangles, theorems on, 41, 53, Exs. 9, 10, 11; 54, Ex. 23; 84, Exs. 7, 10, 11; 246; sextuply, 246 340 INDEX Perspectivity, center of, plane of, axis ot, 36 ; notation for, 57 ; central and axial, 57, between conic and pencil of lines (points), 215 Fieri, M., 95 Hanar field, 55 Planar net, 86 Plane, at infinity, 8, defined, 17; deter- mined uniquely by three noncollinear points, or a point and line, or two in- tersecting lines, 20 ; and line on same three-space are on common point, 22 ; of perspectivity, 36, 75; of points, 55; of lines, 55; equation of, 193, 198 Plane figure, 34 Plane section, 34 Planes, two, on two points -4, B are on all points of line AB, 20, two, on same three-space are on a common line, and conversely, 22 ; three, on a three-space and not on a common line are on a common point, 23 Plucker's line coordinates, 327 Point, at infinity, 8 ; as undefined ele- ment, 15 ; and line determine plane, 17, 20; equation of, 174, 193, 198; of contact of a line with a conic, 112 Point conic, 109 Point figure, 34 Points, three, determine plane, 17, 20 Polar, with respect to triangle, 46; equation of, 181, Ex. 3 ; with respect to two lines, 52, Exs. 3, 5 ; 84, Exs. 7, 9 ; with respect to triangle, theorems on, 54, Ex. 22 ; 84, Exs. 10, 11 ; with respect to n-line, 84, Exs. 13, 14 ; with respect to conic, 120-125, 284, 285 Polar forms, 256 ; with respect to set of tt-points, 256; with respect to regu- lus, 302 ; with respect to linear com- plex, 324 Polar reciprocal figures, 123 Polarity, in planar field, 263, 279, 282, 283 ; in space, 302 j null, 324 Pole, with respect to triangle, 46 ; with respect to two lines, 52, Ex. 3 ; with respect to conic, 120 ; with respect to regulus, 302; with respect to null system, 324 Poncelet, J. V., 29, 36, 58, 119, 123 Problem, degree of, 236, 238 ; algebraic, transcendental, 288; of second de- gree, 245 ; of projectivity, 250, Ex, 14 Product, of two correspondences, 65; of points, 145, 231 Project, a figure from a point, 36 ; an element into, 58, ABC can be pro- jected into A'B'C', 59 Projection, of a figure from a point, 34 Protective collineation, 71 Projective comes, 212, 304 Projective correspondence or transfor- mation, 13, 58 ; general group on line, 08 ; in plane, 268 ; of two- or three- dimensional forms, 71, 152 Projective geometry distinguished from metric, 12 Projective pencils of points on skew lines are axially perspective, 64 Projective pro jectivi ties, 208 Projective space, 97 Projectivity, definition and notation for, 58; ABC-xA'B'C', 59; ABGD^ JSAJDO, 60 ; in one-dimensional forms is the result of two perspectivities, 63 ; if JET (12, 34), then 1234^1243, 82; fundamental theorem of, for linear net, 94 ; fundamental theorem of, for line, 95; assumption of, 95; funda- mental theorem of, for plane, 96 ; for space, 97 ; principle of, 97 ; necessary and sufficient condition for MNAB -r- MNA'B' is Q(MAB, NB'A^, 100; necessary and sufficient condition for MMAB ^ MMA'B' is Q (MAB, MB' A'), 101 ; parabolic, 101 ; ABCD -K-ABDC implies H(AB, CD), 103; nonhomogeneous analytic expression for, 154-157, 206 ; homogeneous ana- lytic expression for, 164; analytic expression for, between points of dif- ferent lines, 167 ; analytic expression for, between pencils in plane, 183; between two conies, 212-216; on conic, 217-221; axis (center) of, on conic, 218; involution belonging to, 226 ; problem of, 250, Ex. 14. Projectivities, commutative, example of, 70 ; on sides of invariant triangle of collineation, 274, 276, Ex. 5 Projector, 35 Properly projective, 97; spatial net is, 97 Quadrangle, complete, 44; quadrangle- quadrilateral configuration, 46; sim- ple, theorem on, 52, Ex. 6 ; complete, and quadrilateral, theorem on, 53, Ex. 8; any complete, may be trans- formed into any other by projective collineation, 74; opposite sides of, meet line in pairs of an involution, 103 ; conies through vertices of, meet line 111 pairs of an involution, 127 ; inscribed in conic, 137, Ex. 11 Quadrangles, if two, correspond so that five pairs of homologous sides meet on a line Z, ,tho sixth pair meets on , 47, perspective, theorem on, 68, Ex. 12 ; if two, have same diagonal triangle, their eight vertices are on conic, 137, Ex. 4 Quadrangular set, 49, 79 ; of lines, 79 ; of planes, 79 INDEX 341 Quadrangular section by transversal of quadrangular set of lines is a quad- rangular set of points, 79 ; of elements protective with quadrangular set is a quadrangular set, 80; Q(MAB, NB'A')is the condition for MNAB-^ MNATK, 100 ; Q(MAB, MR' A') is the condition for MM A B-j- M, M A'B', 101 ; Q(ABC, A'B'V) implies Q(A'B'C', ABC), 101 ; Q(ABC, A'B'C') is the condition that A A', BB', Ware in in- volution, 103 ; Q(P Pa-Po, P, PyPx+v) is necessary and sufficient for P^ + P = P a4 . y , 142; QCP.PoPi, P P,rP,v) is necessary and sufficient for P x P y = P^145 Quadrangularly related, 80 Quadratic binary form, 252; invariant of, 252 Quadratic correspondence, 139, Exs. 22, 24 Quadric spread in S a , 331 Quadric surface, 301 ; degenerate, 308 ; determined by nine points, 311 Quadrilateral, complete, 44 ; if two quad- rilaterals correspond so that five of the lines joining pairs of homologous ver- tices pass through a point P, the line joining the sixth pair of vertices will also pass through P, 40 Quantic, 254 Quaternary forms, 258 Quotient of points, 149 Range, of points, 55 ; of conies, 128-136 Ratio, of points, 149 Rational operations, 149 Rational space, 98 Rationality, net of, on line, 84, 85 ; planar net of, 80-88 ; spatial net of, 89-93 ; domain of, 238 Rationally related, 80, 89 Reducible equation, 239 Reflection, point-line, projective, 223 Reflexive correspondence, 60 Regulus, determined by three lines, 298 ; directrices of, 299 ; generators or rulers of, 299; conjugate, 299; gen- erated by projective ranges or axial pencils, 299 ; generated by projective conies, 304, 307 ; polar system of, 300 ; picture of, 300 ; degenerate cases, 311 ; of a congruence, 318 Related figures, 35 Resultant, of two correspondences, 65 ; equal, 65; of two projectivities is a projectivity, 68 Reye, T., 125, 139 Rohn, K, 309 Salmon, G., 138 Sannia, A., 304 Scale, defined by three points, 141, 231 ; on a conic, 231 Schroter, H., 138, 281 Schur, F., 95 Science, abstract mathematical, 2 ; con- crete application or representation of, 2 Scott, C. A., 203 Section, of figure by plane, 34 ; of plane figure by line, 35 ; conic section, 109 Segre, C., 230 Self -conjugate subgroup, 211 Self -conjugate triangle with respect to conic, 123 Self-polar triangle with respect to conic, 123 Set, synonymous with class, 2 ; quadran- gular, 49, 79; of elements projective with quadrangular set is quadrangu- lar, 80*; harmonic, 80; theorems on harmonic sets, 81 Seven-point, plane section of, 53, Ex. 14 Seydewitz, F., 281 Sheaf of planes, 55 Side, of ft-point, 37 ; false, of complete quadrangle, 44 Similarly placed quadrangles, 60 Simple element of space, 39 Simple n-point, n-line, n-plane, 37 Singly parabolic point, 274 Singular point and line in nonhomoge- neous coordinates, 171 Six-point, plane section of, 54, Ex, 17; in four-space section by three-space, 54, Ex. 24 Skew lines, 24; projective pencils on, are perspective, 105, Ex. 2 ; four, are met by two lines, 250, Ex. 13 Space, analytic projective, 11 ; of three dimensions, 20; theorem of duality for, of three dimensions, 28 ; n-, 30 ; assumption for, of n dimensions, 33 ; as equivalent of three-space, 34; properly or improperly projective, 97; rational, 98; finite, 201, 202; extended, 242 Spatial net, 89, theorems on, 89-92; is properly projective, 97 von Staudt, K. G. C., 14, 95, 125, 141, 151, 158, 100, 286 Steiner, J., 109, 111, 125, 138, 139, 285, 280 Steiner point and line, 138, Ex. 19 Steinitz, E., 261 Sturm, Ch., 129 Sturm, R., 231, 250, 287 Subclass, 2 Subgroup, 68 Subtraction of points, 148 Sum of two points, 141, 231 Surface, algebraic, 259; quadric, 301 Sylvester, J. J., 323 System affected by a correspondence, 65 342 INDEX Tangent, to conic, 112 Tangents to a point conic form a line conic, 116 ; analytic proof, 187 Taylor's theorem, 255 Ternary forms, 258; bilinear, repre- sent correlation in a plane, 267 Tetrahedra, perspective, 43, 44 ; config- uration of perspective, as section of six-point in four-space, 54, Ex. 24: Mbbius, 105, Ex. ; 326, Ex. 9 Tetrahedron, 37; four planes joining line to vertices of, projective with four points of intersection of line with faces, 71, Ex, 5 Three-space, 20; determined uniquely by four points, by a plane and a point, by two nonintersecting lines, 23 ; the- orem of duality for, 28 Throw, definition of, 60 ; algebra of, 141, 157; characteristic, of project! vity, 205 Throws, two, sum and product of, 158 Trace, 35 Transform, of one projectivity by an- other, 208 ; of a group, 209 Transform, to, 58 Transformation, perspective, 18; pro- jective, 13 ; of one-dimensional forms, 58; of two- and three-dimensional forms, 71 Transitive group, 70, 212, Ex. 6 Triangle, 37; diagonal, of quadrangle (quadrilateral), 44; whose sides pass- through three given collinear points and whose vertices are on three given lines, 102, Ex. 2; of reference of system of homogeneous coordinates in plane, 174; invariant, of collinea- tion, relation between projectivities on sides of, 274, 270, Ex. 5 Triangles, perspective, from point arc perspective from line, 41 ; axes of perspectivity of three, in plane per- spective from same point, are con- current, 42, Ex, G; perspective, theo- rems on, 53, Exs. 9, 10, 11 ; 105, Ex. 9 ; 116, 247 ; mutually inscribed and circumscribed, 99; perspective, from two centers, 100, Exs. 1, 2, 3; from four centers, 105, Ex. 8 ; 138, Ex. 18 ; from six centers, 246-248 ; inscribed and circumscribed, 250, Ex. 4 Triple, point, of lines of a quadrangle, 49 ; of points of a quadrangular set, 49 Triple, triangle, of lines of a quadran- gle, 49; of points of a quadrangular set, 49 Triple system, 3 Undefined elements in geometry, 1 United position, 15 Unproved propositions in geometry, 1 Variable, 58, 150 Veblen, 0., 202 Veronese, G., 52, 53 Vertex, of w-points, 36, 37; of n-planes, 37; of fiat pencil, 55; of cone, 109; false, of complete quadrangle, 44 Wiener, H., 65, 95, 230 Zeuthen, H. G., 95 Date Due w Demco 293-5 . S?. r K9!5.M5!!" University Libraries 3 fiHfiS 01111 "Git V39r- V.1 (2) Carnegie Institute of Technology Library Pittsburgh, Pa.